Environment Exploration in Sensing Automation for Habitat Monitoring

Transcription

1 1 Environment Exploration in Sensing Automation for Habitat Monitoring Patrick A. Plonski, Joshua Vaner Hook, Cheng Peng, Narges Noori, an Volkan Isler Abstract We present algorithms for environment exploration in the context of a habitat monitoring task where the goal is to track raio-tagge invasive fish with autonomous surface or groun robots. The first task is navigation aroun an unknown obstacle using input from a front-facing sonar. This capability is important for navigation on inlan lakes because plants an shallow shorelines are har to map in avance. The secon task involves energy harvesting for long term operation. We aress the problem of exploring the solar map of the environment which is use for energy-efficient navigation. For both problems, we present online algorithms an examine their performance using competitive analysis. In competitive analysis, the performance of an online algorithm is compare against the optimal offline algorithm. For obstacle avoiance, the offline algorithm knows the shape of the obstacle. For solar exploration, the offline algorithm knows the geometry of the shaow casting objects. We obtain an O(1) competitive ratio for obstacle avoiance, an an O(log n) competitive ratio for solar exploration, where n is the number of critical points to observe. The strategies for obstacle avoiance are valiate through extensive fiel experiments, an the strategies for exploration are valiate with simulations. Note to Practitioners Most classical automation tasks take place in structure environments such as factories an warehouses. Such environments can be mappe in avance. The maps in turn allow for the offline planning of trajectories. Mapping the environment in avance for automation tasks which take place in large outoor environments can be challenging. Online exploration base on sensor feeback becomes necessary. In this paper, we stuy exploration problems for two sub-tasks of a habitat monitoring application. The first task is navigation aroun unknown obstacles. The secon task is exploration of shaow casting obstacles for solar harvesting. For both problems, we present motion strategies with provable performance guarantees. Combine, these algorithms avance the state of the art towar the goal of monitoring tracking wil-life an other ynamic phenomena with autonomous surface vehicles. I. INTRODUCTION Sensing automation in unstructure environments is becoming increasingly important for many applications incluing home automation where occupancy patterns are monitore for energy management, an farm automation where sensory information is use to guie farm practices. In this paper, we focus on a novel sensing automation task that takes place in a large unstructure environment: habitat monitoring. Since 21, we have been working on builing a robotic sensor network to monitor Common Carp, an invasive fish infesting lakes across the Miwest. Fish biologists an The authors were at the University of Minnesota while this work was performe. 2 Union St. SE KHKH 4-192, Minneapolis, MN, USA, s: {plonski, jvaner, peng175, noori, isler}@cs.umn.eu Preliminary versions of the results presente in this paper appeare in [1] [3]. Fig. 1. (Left) Two robotic boats tracking invasive fish in Minnesota lakes. (Right) Rovers use in winter experiments when the lakes freeze over. watershe management personnel surgically insert raio tags in the fish an manually track them to stuy their behavior an fin aggregations. The tracking process is performe throughout the year by searching for signals emitte from the tags using irectional antennas [4]. This is a ifficult an labor intensive job. The primary goal of our project is to buil a autonomous system which can track raio-tagge fish. We use autonomous surface vehicles in the summer an switch to groun vehicles in the winter when lakes freeze. We have emonstrate that the system is capable of fining tagge fish [5] an localizing them [6], [7]. Yet, moving from proof-of-concept emonstrations to efficient automation remains ifficult. A general challenge associate with sensing automation in large, unstructure environments is ue to the unknown nature of the environment. For example, even though offline maps can be use for planning sensing paths, unmappe obstacles such as plants an shallow waters make navigation ifficult. Similarly, while solar harvesting is a viable option to enable long-term operation, shaows cast by unmappe obstacles such as trees can make harvesting challenging. We stuy algorithmic aspects of environment exploration an make two contributions. First, we investigate obstacle avoiance: how can a robot boat with a sonar navigate aroun an unknown obstacle most efficiently. We formulate two problems for ifferent restrictions on the properties of the obstacle. Secon, we investigate the problem of exploring a solar map to learn the heights of shaow casting objects (e.g. trees on a shoreline). Once these heights are known, the map of energy gains can be compute for any time of ay, an energy-efficient paths can be planne. We provie extensive fiel experiments justifying our obstacle avoiance approach, an for both problems we prove competitive ratios between our online strategies an the optimal offline solutions. The rest of the paper is organize as follows. In Section II

2 Data LS cubic RANSAC SINE Fig. 2. (Left) The tag which is implante in fish. The tag is approximately 5 cm in length, with a 3cm antenna trailing off (2 inches, 12 inches). (Mile) The antenna is approximately 61 cm in iameter (24 inches). The tags transmit an uncoe raio pulse once per secon in the 47.9 MHz range. The antenna is irection-sensitive, meaning the perceive strength of the signal is epenent on the orientation of the antenna relative to the irection to the tag. In this way, it is possible to estimate the bearing to the transmitting tag. (Right) A coarse sampling of the signal strength. The horizontal axis is the pan angle an the vertical is the measure signal strength. Least squares estimation of a sine curve typically provies the best estimates. we present the etails of our carp tracking system. In Section III we iscuss relevant literature. In Section IV we provie an introuction to online competitive strategies. In Section V we present our strategies for obstacle avoiance. In Section VI we present our strategies for solar map exploration. Finally, Section VII is the conclusion where we iscuss future work. II. SYSTEM OVERVIEW We execute our carp tracking algorithms on robot boats in the summer an robot rovers in the winter which rive along the frozen surface of the lake. In this section we will present the etails of these systems. Our boats are built on the 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. Our rovers are built on the Clearpath Husky A1 an A2 2, which are both rugge an heavy ski-steer vehicles. The A1 is.86m long by.65m wie with a atasheet mass of 35kg an a maximum payloa of 4kg. The A2 is.99m long by.67m wie with a atasheet mass of 5kg an a maximum payloa of 75kg. Both rovers can travel at 1 m/s. The rovers are irectly controlle with a serial port connection from a laptop computer. The boats are controlle with Aruino Mega256 microcontroller boars that communicate with the motor controllers, servos, an sensors, an are in turn commane by laptop computers over irect Ethernet using UDP. Self-localization is achieve through the use of a GPS unit an a compass, filtere with an Extene Kalman Filter. On the rovers, wheel oometry is use as well. All software is compatible with the Robot Operating System. Our boats can be configure with a forwar-looking, singlebeam sonar unit was mounte on the bow of the vehiclethe sonar unit is an Imagenex 852 Digital Echo Souner 3. 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. [8] 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. The boats an rovers can also be equippe with solar panels to increase their system lifetime. The resulting system must have an ongoing estimate of expecte solar power over the working environment, an inclue this information as part of a global path planner, as shown in [9]. III. RELATED WORK In this section we will first present the state of the literature for sonar-base obstacle avoiance. Then we will present the state of the literature for solar-aware path planning, as it relates to our exploration problem. Obstacle avoiance for Autonomous Surface Vehicles (ASVs) commonly focus on obstacles that can be etecte from above the water, using raar [1] or liar [11], [12]. Although sonar-powere obstacle avoiance is more common for unerwater vehicles than it is for surface vehicles, most obstacles have a below-water presence, an in shallow waters many non-traversable regions can be only etecte from sonar [8]. Sonar sensors are also much less expensive than laser scanners or raar, an easier to use than vision. Heiarsson an Sukhatme [8] equippe their surface vehicle with a singlebeam sonar mounte on a mechanical evice which allowe rotation for scanning. They accomplishe obstacle avoiance using Vector Fiel Histograms on the sonar returns. Most obstacle avoiance in sonar-equippe vehicles (e.g. [13], [14]), is accomplishe with Artificial Potential Fiels [15], where etecte obstacles exert a virtual repulsive force to balance the virtual attractive force exerte by the goal position. The basic APF approach is easy to implement, but it is not robust in cluttere environments or when obstacles have complex shapes. Local minima are a particular problem. The Vector Fiel Histogram [16] improves robustness by allowing a probabilistic estimate of obstacle occupancy, using a certainty gri (this was the potential fiel metho of choice for [8]. The successor to the VFH, known as VFH+ [17], improves stability in part by explicitly compensating for the

3 3 robot with an movement moel. However, as it is still a local planner, a robot equippe with VFH+ can steer towars known ea ens. Usually vehicles are equippe with multibeam sonar, however Calao et al. [18] use Histogramic In-Motion Mapping (HIMM) [19] to buil gri maps of the sense obstacle from a single beam sonar. This allowe the system to be constructe at greatly reuce cost. A convex polygon obstacle was built base on the line segments extracte by Hough transform. Given the constructe map, a potential fiel metho was use for navigation. There are alternate approaches to obstacle avoiance besies vector fiel methos. Petres et al. [2] presente a metho of path planning for AUVs which assume full knowlege of the environment. Similarly, Eichhorn [21] propose a complex planning metho for an unerwater glier which use a prior map of the environment, augmente with on-the-fly sonar reaings to avoi obstacles as they were encountere, while still steering towars the estination. The reactive control metho use for obstacle avoiance was base on artificial graients, which are very similar to the more common APF techniques [22]. Petillot et al. [23] mappe the surrouning obstacles by segmenting an extracting obstacle features from the sonar image. Then vehicle motion planning was converte into a nonlinear programming problem while the extracte obstacles were treate as inequality constraints [24]. Nonlinear programming approaches can be less susceptible to local minima than vector fiel methos. Khorrami an Krishnamurthy [25] propose a sophisticate optimization metho on the current best environment estimate. Kawano [26] propose a metho which plans a Markov Decision Process (MDP) policy to control an AUV in the presence of obstacles an unknown currents. These methos can hanle more information than vector fiels, but they are all only guarantee to be goo with respect to the observations mae so far. They provie no guarantees when the obstacles must be observe on-the-fly, as in our scenario. Existing techniques for sonar-base obstacle avoiance provie no guarantee on the length of the final robot trajectory; an early turn in the wrong irection can result in a much longer path than necessary. This is perhaps more of an issue for surface vehicles than the unerwater vehicles most commonly stuie, because of the reuce imensionality of their omain. In this work we present a metho which can hanle arbitrarily large obstacles an still reach the target in finite time with a near optimal competitive ratio. When planning long term missions for solar-powere robots, the amount of energy harvesting nees to be consiere to ensure that the system batteries o not run out of charge. In open environments, such as in Antarctica [27], on the open ocean [28], or in the sky [29], the energy epens only on the sun position an the orientation of the vehicle (an weather conitions). However, in cluttere groun environments, such as in urban or foreste areas, shaows have a consierable effect on energy harvesting via photovoltaic panels. Several authors have presente methos of controlling vehicles equippe with solar photovoltaic panels in ways that o not require a prior map. Vaussar et al. [3] use a simple source seeking approach to maximize the lifetime of a groun vehicle in a cluttere inoor environment. Hartono et al. use a neural network to allow for on-the-fly learning an seeking of sun positions an timing, without any formal map construction. Finally, the Solar Chemical Detection Robot [31] is teleoperate, so its human operator can compensate for any variability in energy collection. These methos have few guarantees on their performance. In complex environments, the best paths are obtaine through knowlege of a full solar map of the environment. The TEMPEST mission-level path planner [32] uses the known position of the sun together with known nearby terrain to perform raytracing an compute a shaow map, which is then use to compute the net energy of any potential path. Recently Kaplan et al. [33] presente a particle swarm base metho which use a prior solar map to plan time-optimal paths for a solar powere groun vehicle, subject to a power constraint. Given that sensing the terrain in high etail is a ifficult task, in [9] an [34] we examine the problem of learning the solar map using only measurements of position an solar current associate with positions an times. This technique relie on having measure a suitable coverage of the environment. In this work we look to ensure this is the case. IV. PRELIMINARIES Analyzing a movement strategy can be challenging when we on t know what the environment looks like. A common metho in the literature is to look at the competitive ratio between the online strategy an an optimal offline strategy with full information. 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 an solar exploration problems stuie in this paper are online problems since we must choose a motion strategy that reacts to the measurements receive uring execution without knowing the exact geometry of the environment in avance. The performance of an online algorithm A is measure using its competitive ratio which is given by c(a) = max σ A(σ) A (σ) where σ varies across all inputs, A(σ) is the performance of A for input σ an A (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. Some recent examples of exploration strategies analyze using competitive ratios inclue [35], [36], an an overview of online competitive analysis can be foun in [37]. 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

4 4 is formulate as follows: The cow an the gate are on a line, the gate s location is unknown, an the cow starts 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 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 fori 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. 7). Using elementary computations, it can be shown that the oubling strategy has a competitive ratio of 9 [38]. Due to sensing an motion uncertainty an constraints, esigning a similar online algorithm for obstacle avoiance is non-trivial. In this paper, we first present novel online strategies for obstacle avoiance, analyze their competitive ratios an valiate them in fiel experiments. Then, we consier a relate solar map exploration problem where there are n points to fin, but they can be foun in any orer, an our sensor gives information on the irection. We present strategies for this problem, an show how to obtain a competitive ratio which is logarithmic in n. V. OBSTACLE AVOIDANCE 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 single beam sonar (Figure 3). The problem is now more complicate because of such factors as the narrow with of the sonar, the noisy an sometimes ambiguous sonar reaings, an the motion constraints of the boat. Fig. 3. The autonomous boat with sonar visible In the literature, there are two primary approaches in solving this type of online navigation problems. BUG Algorithms [39] 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 [4] [42]. However, these algorithms are rarely implemente on real Fig. 4. 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. 5. 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 traveling towar line l 2, up to istance s, or returning to l if it has encountere the obstacle. Right: An example execution of AvanceRetreat. 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. robot systems since they o not aress motion an sensing constraints. In this section, we present novel navigation strategies which are both provably efficient an suitable for fiel implementation. We first start 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 orientation 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 IV) 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.

5 5 A. Problem Formulation 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 4. Motion Constraints: The ASV can position its ruer to irect thrust from a single motor. The maximum angle of the ruer etermines a minimum turning raius R t, which is about 5m in our system. The forwar velocity can be set inepenently of the ruer, so we consier it as fixe. Therefore time to complete any feasible trajectory is proportional to its length. 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 start 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. Problem 1 (Rectangular Obstacle). Given a rectangle R, a line l containing 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 4. 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 5. In Section V-C, 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. B. Strategy for Problem 1 an 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 Fig. 6. 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 U 2u 1 = 2 R 2 t +D2 t 2. Fig. 7. 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. 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 6. 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 U 2 R 2 t +D 2 t 2, when the maneuver is starte at istance t+r t, as shown in Figure 6. This means t is the istance between the center of the Sweep circle an the obstacle. 4) GoTo follows a path of length equal to the Eucliean istance between two points. 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

6 en y position (meters) 5 start y position (meters) 5 en y position (meters) 5 en start start x position (meters) x position (meters) 5 1 x position (meters) (a) (b) (c) () Fig. 8. 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 U = 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. In (b) an (c), 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. In (), 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. 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 IV, the robot will GoTo waypoints which alternate left an right from x, such that each point x i is at position U( 2) i 1. At each of these points, incluing the first etection point, the robot will execute a Sweep, 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 travele uring all Go- To 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. 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., ǫ). Note, the lost-cow algorithm will arrive at the point while traveling no farther than 9 [38]. 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 ǫ 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 U. 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 = U 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 U 4 n 1 U 4 n, which implies n = log U steps are require on the positive sie, or at most n = 2 log 4 U 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 Rt U 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 U. 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 +

7 7 4πR t log 4 U L + L L. Since < L, c(circlesweep) 13+ 4πR t log 4 U U log = 13+4πR 4 U t U Also note log b x x = 13+4π R t U log 4 U U 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. 1) Fiel Experiments: For our first fiel experiments we execute CircleSweep at Lake Staring, Minnesota, USA. We use the north 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 U = 25m; this is reasonable if D 15.21m an t 1m. 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 8. 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. C. Strategy for Problem 2 an 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 parameterize by two parallel lines l an l 2 which are inferre base on the assume with an orientation of the obstacle. The safety line l is 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 an l 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 5. The optimal path, starting at position x, oes not enter the convex hull of the polygon. It passes through a corner point a 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 starting 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. Upon returning, the robot will move to a location which is istance U ( 2) i 1 along the line l an repeat the process (see Figure 5 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, U can be represente by the beam with, an a non-etection is simply no echo returns. Secon, we assume that the robot can always turn aroun to return to the line, thus the 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. Lemma 3 (Number of Probes). During an execution of AvanceRetreat, the robot makes 2 log 4 U 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 fromlto the object plus the cost to make four quarter turns. Without loss of generality, let be the point onl 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. Using the same analysis as Lemma 2, we obtain an upper boun of log 4 U probe steps require on the positive sie, or 2 log 4 U probes require in total. Theorem 2 (Cost of AvanceRetreat). Avance- Retreat has a competitive ratio that is Θ(log U ). Proof. We know the robot travels no more than istance 12+ (2s+2πR t ) log 4 U, by combining the travel (Lemma 1) an

8 en start 5 5 y position (meters) 5 start y position (meters) 5 y position (meters) 1 15 start y position (meters) 1 15 start 1 1 en en en x position (meters) x position (meters) x position (meters) x position (meters) (a) (b) (c) () Fig. 9. 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, 1) for the secon two. The crosses give the orientation of the safety line l, an they are space apart at the unit istance U = 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 trials (a) an (b), the experiment was terminate manually after the boat successfully etecte the shoreline in four ifferent probes. In trials (c) an (), the boat successfully navigate aroun the shoreline to the other sie of l 2 in its fifth probe. probe (Lemma 3) steps. Note that the optimal path is at least length 2 +s 2, as illustrate in Figure 5. Thus, c(avanceretreat) which proves the theorem statement. = 12+(4s+4πR t) log 4 U 2 +s 2 = s + (4s+4πR t) log 4 U 2 2 +s 2 ( 12+ log πR ) t, U 2 +s 2 Note that the part of the competitive ratio that grows logarithmically 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. 1) Fiel Experiments: For our secon fiel experiments we execute AvanceRetreat 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 U = 25m. The paths followe by the robot, along with the estimate positions of all etecte sonar objects, are shown in Figure 9. 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. VI. SOLAR EXPLORATION Limite battery life is a major ifficulty face by present sensing automation systems in the wil. Previously the utility of solar-aware path planning has been emonstrate. In [9] an [34] we constructe solar maps using only previous measurements of solar power associate with positions, an use the maps to plan energy-aware paths. The methos implicitly relie on the existence of prior solar measurements in the region where the the robot was to travel. In this section we aress the exploration component necessary to generate an accurate solar map. Specifically, we are intereste in visiting all of the critical points which mark the tops of the shaows cast by n known objects. We assume that we know the positions of the objects, e.g. from satellite imagery, or because of a known shoreline. However, we on t know where the critical points are, because we on t know the object heights. Any coverage pattern or ranom walk will eventually make the necessary observations, however when a robot executes these strategies it might travel a great istance. We woul like some metho to ensure that we expen the least energy an time in the exploration step. To this en, we use a quatreebase exploration strategy which uses observations of sun an shae to refine its estimate of the heights of shaow casting

9 9 Fig. 1. Here is an example of our problem setup. Each object has known position on the plane but unknown height. To fin the height the corresponing site in S must be visite. The positions of S are unknown to the robot, however a boun on the height of each object constrains its corresponing site to lie on a line segment parallel to the sun. objects in the environment. We emonstrate that this strategy obtains a boune worst-case competitive ratio between the istance travele using our exploration algorithm an the istance travele with the optimal algorithm. The boun is logarithmic in the number of critical points that must be observe. A. Problem Statement We formalize the problem of fully exploring the shaow map in the following manner: We have n objects in our environment with known position but unknown height. To be able to claim that we have fully explore the environment, we nee to precisely etermine each of these n heights. From the heights we will be able to obtain the shaow map for any time of ay. We know the angle of the sun at any moment (see [43]), so to fin the height of an object it suffices to fin the length of its shaow. This is equivalent to fining the critical point where a solar panel will switch from collecting irect insolation to only collecting iffuse insolation. We call these critical points sites {s 1,s 2,...,s n } = S. We assume there are upper an lower bouns on the object heights, so each site therefore lies on a known line segment. We also assume that uring the exploration step the sun oes not appreciably change position, thus all line segments are parallel. Without loss of generality, therefore, we perform a coorinate transform base on the angle of the sun. The x positions of the sites, {s x 1,sx 2,...,sx n } = Sx, are known a priori, an it is only the y positions, {s y 1,sy 2,...,sy n } = Sy that are uncertain. However, there is an initial estimate of the range that each s y i lies in, enote yi u an yi l for the upper an lower limits, respectively. Without loss of generality, s x i Sx,s x i sx i 1. See Figure 1 for an example problem setup for an environment with three objects. We make some simplifying assumptions: We assume in this section that our exploring robot r is a holonomic point robot. We are primarily intereste in the case where r begins outsie the x range, so we fix the initial x coorinate of r at an require that all S x are positive. As r explores the environment, it learns more about the true positions of S. We assume that walking farther away from an object than a critical point will result in sun, an walking closer will result in shae; (s x i S x ) = r x, the robot learns whether s y i or r y is greater, by measuring the solar insolation on its panel. This problem iffers from the problem presente in Section V in that now we can observe which irection we nee to travel to reach the target, but now there are n critical points that nee to be reache, instea of a single obstacle bounary. However, unlike in Section V, we o not consier the motion constraints of the robot. This is equivalent to assuming that the turning raius is small compare with the size of the shaows. Our aim is to fin an algorithm for r that explores the environment an visits all ofs. We o not require that r return back to the starting position; the mission ens when the last site is visite. To juge the quality of our exploration algorithm we o not assume any probability istribution but rather consier the worst case competitive ratio between the istance r travels when commane by our algorithm, l(r), an the istance r woul travel were it commane by an optimal offline algorithm that simply solves the traveling salesman path for the starting position of r an the true positions of the sites. We enote the TSP solution as R, an we can formally state now that our goal is to minimize the following competitive ratio: ( ) l(r) c(r) = max S l(r ) In this work we present an exploration algorithm for r an emonstrate that its worst case competitive ratio is logarithmic in the number of objects in the environment n. B. X-Sweep Strategy Before we introuce our full algorithm we first present a basic naíve strategy which has poor worst-case performance but which we will later use as a subroutine. The X-Sweep strategy simply visits all the input line segments in orer from lowest x-coorinate to highest. It goes to the closest point on the segment first an walks along it until it reaches the site, then moves on until it has visite every site. Clearly if there is not much ifference between max(s y ) an min(s y ), the X-Sweep Strategy will perform close to optimal. However, it is easy to construct a situation where X-Sweep performs poorly. Suppose the s y j Sy alternate between an some large value m, an furthermore suppose = x n x is very small compare with m. In this case the optimal strategy cuts along y =, then cuts back along y = m, with a total path length close to m. In comparison, X- Sweep walks up an own the line segments an its total path length is close to nm. Therefore the worst case competitive ratio for X-Sweep is at least as ba as O(n). C. QuaExplore Strategy We saw that the X-Sweep strategy can have a competitive ratio as ba as O(n). This is because it can backtrack very far with each walk along a segment. To obtain a better boun we nee a metho to balance walking along the segments with cutting across the segments. One such metho is our Qua- Explore strategy: it cuts across segments when they are long an executes X-sweep once the belief line segments are short enough that performing X-sweep is guarantee inexpensive.

10 1 Fig. 11. Here is an example Q constructe by QuaExplore for an environment that has 8 sites. Q is shown here separate by level but it is constructe in epth first fashion, with chilren explore in counter-clockwise orer from the upper right. At each noe it is etermine which of the four caniate squares A contain sites; those that contain sites become squares for chilren. If the square for a noe only contains a single site the noe executes the leaf strategy an creates no chilren. This leaf strategy is also execute whenever the maximum epth h(q) is reache. 1) Description: Here is the escription of Qua-Explore. First we consier the case where = x n x 1 = max(y u ) min(y l ). That is, the arena is square. QuaExplore is base on constructing a recursive quatree structure that we call Q. See Figure 11 for an example of the structure of Q. The structure is constructe as it is traverse, epth first. Each noe q i of Q consists of a square in the plane A(q i ), a pointer to the parent, an pointers to the possibly four chilren {c 1,...} = C(q i ), C 4. Noes are numbere as they are create, an q correspons with the entire arena. In aition to Q the robot must maintain knowlege about its position, an its belief Y l an Y u on the lowest an highest possible values for S y consistent with the initial belief an the online observations. For each q i the algorithm generates four caniate squares {a 1,a 2,a 3,a 4 } = A(q i ) by evenly iviing A(q i ). Those caniate squares which contain sites will become associate with new noes that are chilren of q i, an control will be passe to them, counter-clockwise, in turn, before returning up to the parent of q i. We fix the maximum epth of Q as h(q) = log 2 (n) so that the ege length of a leaf square is proportional to /n. We execute a leaf strategy whenever this epth is reache. Also, we execute the same leaf strategy if at any time there is a q that only contains one site. See Figure 12 for an example of our strategy for a leaf noe an a non-leaf noe. Actions commane by a leaf noe q i : The leaf noe actions are functionally almost ientical to performing the X- sweep strategy with A(q i ) as the arena. From the leftmost or rightmost borer ofa(q i ), rive to a pointp on the linex = s x j for whichever s j A(q i ) has the least x istance from r x. The y value of p is selecte such that yj l py yj u an the istance between p y an r y is minimize. Then walk along the line segment of s j until s j is visite, an then move on to the next closest s x k s k A(q i ). Repeat this process until every site in A(q i ) has been visite. Then pass control back to the parent by visiting the intermeiate y of A(q i ) at the opposite x ege from the start of r q i. Actions commane by a non-leaf noe q i : Perform a y- cut by riving r along the line segment that splits the upper caniate subsquares {a 1,a 2 } A(q i ) from the lower ones {a 3,a 4 } A(q i ). Ajust all Y u an Y l that pass through the cut. It is clear at this point which of A(q i ) contain sites, so associate these squares with chilren. For chil j, q i is responsible for elivering r to the mile y of a j, at either Fig. 12. Left: An example execution of the level 1 strategy when max(s y ) min(s y ) >. The starting position (,) is on the lower left an r follows the thick soli gray path, in the orer given by the arrows. When it is certain there is a site in one of the subsquares a i A(q ), a chil is create for that subsquare an control is passe to the chil at the start of the ashe gray line. Control is resume by q at the other en of the otte line. Mile: The QuaExplore strategy for a noe q i that is not a leaf. After performing the cut from left to right it becomes clear that all sites in the noe exist in a 1 an a 3, the upper right an lower left caniate subsquares, thus C(q i ) = 2. Assuming that the next level is the maximum epth h(q), each of the two chilren must perform the leaf strategy. Right: The leaf strategy for c 1 C(q i ) is performe right to left, an it simply performs X-sweep an visits all of the sites in a 1 in orer. the left ege or the right ege of a j (right ege for a 1 an a 2, left ege for a 3 an a 4 ). Each chil will return control at the opposite sie of its square. After the last chil returns control, q i passes control back to its parent by returning r back to the ening point of the y-cut. Now, consier the case where the arena is not square. If the arena is longer than it is tall, we simply take any square with ege length that contains the initial belief line segments as A(q ) in the algorithm. If the arena is taller than it is long, we nee to introuce a special strategy only use by q. The ege lengths of the chilren of q in this case are all, so to remain consistent the chilren are efine as level an the special q is efine as level 1. Actions commane by an initial q that has a non-square corresponing region: Perform the first y-cut at y = an ajust Y base on the observations. If afterwars there is any y u i >,y u i Y u, perform the next y-cut at y =. All cuts in q will occur at y = k where k is an integer. The orer of cuts is: first increment k until max(s y ) is iscovere, an then ecrement k from until min(s y ) is iscovere. All subsquares are boune above an below by instances of y = k. When after any cut it becomes certain there is one or more sites in a subsquare, a chil is create for the subsquare an control is passe to the chil at the usual extreme x,