Inspection Planning for Sensor Coverage of 3D Marine Structures

Brendan Englot and Franz Hover, Member, IEEE

Abstract— We introduce an algorithm to achieve completesensor coverage of complex, three-dimensional structures sur-veyed by an autonomous agent with multiple degrees offreedom. Motivated by the application of an ocean vehicleperforming an autonomous ship hull inspection, we considera planning problem for a fully-actuated, six degree-of-freedomhovering AUV using a bathymetry sonar to inspect the complexstructures underneath a ship hull. We consider a discrete modelof the structure to be inspected, requiring only that the model beprovided in the form of a closed triangular mesh. A dense graphof feasible paths is constructed in the robot’s configurationspace until the set of edges in the graph allows completecoverage of the structure. Then, we approximate the minimum-cost closed walk along the graph which observes 100% of thestructure. We emphasize the embedding of observations withinthe edges of the graph as a means of utilizing all availablesensor data in planning the inspection.

I. INTRODUCTION

There are numerous practical applications requiring peri-odic or persistant surveillance of the indoor/outdoor envi-ronment and structures within the environment. To automatethese tasks, a variety of planning algorithms have beendeveloped to solve the robot coverage problem. For applica-tions in 2D workspaces (e.g., demining, snow plowing, lawnmowing) in which the agent possesses relatively few degreesof freedom, a cellular decomposition of the free space istypically sufficient to achieve coverage [1]. Approaches inwhich the inspection of a structure is the goal have oftenmodeled the task as an “art gallery problem”, which requirescomputation of the minimum number of guards who cantogether observe the entire gallery [2]. In addition, some cov-erage algorithms have implemented planning strategies basedon prior knowledge of the specific geometry of the structurebeing covered, as in car painting [3] and building inspections[4]. Finally, various challenging issues remain unsolved froma computational geometry perspective if highly accurate 3Dmodels are a desired product of the inspection, especially inthe absence of a priori knowledge [5].

Our application of interest is the inspection of a shiphull by an autonomous underwater vehicle (AUV), in whichacoustic sensing is required to characterize the surroundingenvironment and complete sensor coverage is necessary tolocate foreign objects that may have been placed on the hull.A fully-actuated, hovering AUV has been developed for theinspection task, equipped with imaging and bathymetry sonarto perform the inspection and an inertial measurement unit

This work was supported by the Office of Naval Research under GrantN00014-06-10043, monitored by Dr. T.F. Swean

B. Englot and F. Hover are with the Department of Mechanical Engineer-ing, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cam-bridge MA 02139 USA benglot@mit.edu hover@mit.edu

(IMU) and Doppler velocimetry log (DVL) for underwaternavigation [6]. Efforts are also underway to use the sonardata itself to improve navigation via simultaneous localiza-tion and mapping [7]. With respect to the motion planningproblem, a large portion of the hull is relatively flat andcan be inspected suitably by an imaging sonar and a planarvehicle trajectory, which a cellular decomposition approachwould solve easily. It is the complex structures under thehull, such as the propellers, rudders, and portions with severecurvature, which require nontrivial inspection paths and theuse of bathymetry sonar to accurately observe their geome-tries. To model these structures a discrete representation ofthe ship hull will be used, allowing our algorithm to utilizea CAD model or a preliminary range scan (such as thebathymetry scan depicted in Figure 1) for the planning task.In addition, five degrees of freedom are available for planningthe inspection (surge, sway, heave, yaw, and sonar pitch).

Given the dimensionality of the problem, a sampling-basedmotion planning algorithm, in which robot configurationsare sampled quasi-randomly and used to catalog feasiblepaths through the workspace, is an appealing alternative toan exhaustive deterministic search of a multi-dimensionalconfiguration space (C-Space). Prior work in sampling-basedplanning of inspection paths has solved the problem byadopting the aforementioned “art gallery” approach of choos-ing a set of configurations which offer complete coverage andfinding a feasible tour among these nodes [8], [9], [10]. Thisapproach plans using a graph in which sensor informationis embedded in the nodes and the edges are used as ameans of transit between nodes. We propose instead to usethe full set of sensor information obtained from traversingthe edges of the graph, modeling the inspection task as apostman problem rather than a traveling salesman problem.The postman framework has been used previously to solvedeterministically-designed inspections for point robots in a2D workspace [11], [12], in which the robot possesses asmall field of view relative to the size of the workspace. Weadopt this information-efficient approach both to accomodatethe small sensor footprint of the bathymetry sonar and toensure that all views of the structure are utilized in planningthe inspection.

In Section II we introduce the algorithm used to constructa dense graph whose traversal is guaranteed to offer completesensor coverage of the structure of interest. In Section IIIthe method for finding a minimal-cost closed walk on thegraph which offers complete sensor coverage is presented.A closed walk is sought to allow cyclical traversal of theinspection path for persistent surveillance. Section IV offersresults of the algorithm using a simulation of the Bluefin-

The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems October 18-22, 2010, Taipei, Taiwan

978-1-4244-6676-4/10/$25.00 ©2010 IEEE 4412

Fig. 1. Imaging and bathymetry sonar data from a survey of a 65’boat are depicted. Included is a photograph of the Bluefin-MIT HAUV,version 1B, which was used to perform the survey. Imaging sonar mosaiccourtesy of AcousticView. Bathymetry sonar reconstruction courtesy ofHordur Johannsson, MIT.

MIT Hovering Autonomous Underwater Vehicle (HAUV)and triangle meshes of the marine structures of interest toour application.

II. GRAPH CONSTRUCTION ALGORITHM

Given a triangle mesh which represents the structure to beinspected and the obstacles in the surrounding environment,an iterative algorithm is used to construct a dense graph offeasible configurations (nodes) and feasible paths betweenthem (edges). Although this approach draws inspiration fromthe well-known Probablistic Roadmap algorithm (PRM) [13],some distinctly different design choices are made to developa dense graph instead of an expansive, sparse graph.

We denote a robot configuration by q ∈ ℜN , where N isthe dimension of the configuration space, and a discrete nodeof the structure mesh by s ∈ ℜ3. Given a feasible initialconfiguration q0, the set of mesh nodes in the structure,referred to hereafter as Structure, and the set of trianglesbelonging to the structure, Algorithm 1, BUILD GRAPH,begins constructing a graph in configuration space usingADD TO GRAPH(), which is described by Algorithm 2.Configurations qrand are sampled quasi-randomly (using aSobol sequence) and Algorithm 4, FIND NEIGHBORS(),is used to connect qrand to a user-designated number ofneighboring configurations which are already on the graph.

Taking qrand as input, FIND NEIGHBORS() drawsstraight-line paths through C-Space between qrand and thedesignated number of nearest neighbors. Each candidate edgeis then checked for collisions with obstacles, and all nodess in the structure mesh are checked by OBSERVAT ION()to determine whether they lie inside the workspace vol-ume swept by the robot sensor along the traversal of eachcollision-free candidate edge. This swept volume will differdepending on the constraints of the robot, and specificallyin the case of the HAUV we define a sequence of steps bywhich the agent travels from one configuration to another.The user may allow the agent to change its position andorientation simultaneously, but for the purpose of obtaininga swept volume which is easy compute and to implement

during the underwater inspection we first require the agentto adjust its orientation to that of the adjacent node, andthen to move to the position of the node in a separate step.In addition, the swept volume may vary depending on thedirection of travel along the edge, and if this is the casethe observations along both directions of travel must becataloged by OBSERVAT ION().

For all s that lie within the sensor’s swept volume,OBSERVAT ION() proceeds to check the line of sight be-tween the mesh node s and the robot configuration alongthe candidate edge which sees s. This is achieved using aray-tracing algorithm which checks for intersection betweenthe line of sight and the triangles of the structure mesh. Allstructure nodes s that are observed by an edge of the graphare cataloged, and any structure nodes being observed forthe first time are removed from the set Unobserved. Afterthe observations of collision-free edges are checked, thecollision-free edges are returned by FIND NEIGHBORS().

Once the size of Unobserved is diminished to acertain percentage ε of the size of Structure, Algo-rithm 1 begins calling ADD MISSING V IEW () instead ofADD TO GRAPH(). ADD MISSING V IEW (), which is de-scribed by Algorithm 3, is designed to locate the remaininghandful of structure nodes that have not been seen, once thevast majority of the structure has already been observed bythe graph. It does so using NEWTON SEARCH(), whichapplies Newton’s method for root-finding using the robot’sJacobian pseduoinverse to find a configuration qview whichsees the missing structure node smissing. The configurationqview returned by NEWTON SEARCH() is planted at thecenter of a new edge in C-Space, and if the swept volume ofthe edge sees a member of Unobserved, the edge’s endpointsq1

view and q2view are subjected to the same collision check and

observation check used in FIND NEIGHBORS(). To ensurethat a consistent diversity of good configurations is returnedby NEWTON SEARCH(), every call to this algorithm isinitialized with a randomly sampled configuration qrand .

After the entirety of the discrete structure has been ob-served by the graph’s edges, graph construction stops and aclosed walk is found which selects an efficient subset of thegraph to use as an inspection path.

III. PATH-FINDING ALGORITHMA. Exact Formulation for the Optimal Inspection Path

First we present the integer programming formulationwhich solves exactly for the minimum-cost subtour thatobserves the entire structure. The graph of feasible pathsconstructed in C-Space is treated as a flow network, in whicha unit flow represents a single traversal of an edge by therobot, and each edge of the C-Space graph is modeled as apair of directed edges. Associated with each directed edge isthe set of observations gathered by the agent along the edge,which in some cases may be the same for both edges inthe pair, and in other cases each member of the pair maycollect different information (depending on the geometryof the robot’s sensor and its kinematic constraints). Theconstraints of the graph are enforced on the flow variables

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Algorithm 1 BUILD GRAPH(q0,Structure)1: G.init(q0)Unobserved← Structure2: while Unobserved 6= /0 do3: if (Unobserved.size()/Structure.size())< ε then4: ADD MISSING V IEW (G)5: else6: ADD TO GRAPH(G)7: end if8: end while9: return G

Algorithm 2 ADD TO GRAPH(G)

1: FeasibleNeighbors← /02: while FeasibleNeighbors = /0 do3: qrand ← DRAW SAMPLE()4: FeasibleNeighbors← FIND NEIGHBORS(qrand)5: if FeasibleNeighbors 6= /0 then6: G.addNode(qrand)7: for qi ∈ FeasibleNeighbors do8: G.addEdge(qrand ,qi)9: end for

10: end if11: end while

through the incidence matrix A, which contains a row foreach node in the graph and a column for each flow variable,with Ai j ∈ {0,1,−1}. The observations made along the graphare enforced on the flow variables by the constraint matrix B,which contains a row for each flow variable x and a columnfor each discrete node s in the structure being inspected.Entry B jk is 1 if edge j observes mesh node k, and 0otherwise. Using this modeling framework, the problem isstated by the following:

minimize ∑j

c jx j

sub ject to ∑j

Ai jx j = 0 ∀ i

∑j

B jkx j ≥ 1 ∀ k

∑j(A1 j)

2x j ≥ 2

x j integer ∀ j (1)

The indices i, j, and k refer to the graph’s nodes, flowvariables, and the discrete structure nodes, respectively. Thecost ci j applied to each flow variable may represent the timeor energy consumed by the agent while traversing the edgerepresented by flow variable x j. The first constraint enforcesthe structure of the C-space graph, the second requires thatevery node in the structure is observed by the agent at leastonce, and the third requires that part of the closed walk mustpass through the node on which the graph was initialized(designated i = 1), which is assumed to be the configurationfrom which the agent starts the inspection.

In its current form, (1) may allow the minimum-cost flow

Algorithm 3 ADD MISSING V IEW (G)

1: f oundMissingView← f alse2: FeasibleNeighbors1← /0, FeasibleNeighbors2← /03: smissing←Unobserved.getFirst()4: while f oundMissingView = f alse do5: qrand ← DRAW SAMPLE()6: qview← NEWTON SEARCH(qrand ,smissing)7: {q1

view, q2view}← DRAW EDGE(qview)

8: if FIND V IEW (q1view, q2

view) = true then9: FeasibleNeighbors1← FIND NEIGHBORS(q1

view)10: FeasibleNeighbors2← FIND NEIGHBORS(q2

view)11: if FeasibleNeighborsi 6= /0 ∀ i ∈ {1,2} then12: f oundMissingView← true13: G.addEdge(q1

view, q2view)

14: for qiview ∈ {q1

view, q2view} do

15: G.addNode(qiview)

16: for q j ∈ FeasibleNeighborsi do17: G.addEdge(qi

view,q j)18: end for19: end for20: end if21: end if22: end while

Algorithm 4 FIND NEIGHBORS(qrand)

1: FeasibleNeighbors← NEAREST NEIGHBORS(qrand)2: for qi ∈ FeasibleNeighbors do3: if COLLISION(qrand ,qi) = f alse then4: for s j ∈ Structure do5: if OBSERVAT ION(qrand ,qi,s j) = true then6: Unobserved←Unobserved\{s j}7: end if8: end for9: else

10: FeasibleNeighbors← FeasibleNeighbors\{qi}11: end if12: end for

to consist of several unattached closed walks on the graphrather than a single, continous closed walk. To eliminatethis possibility, additional constraints must be included inthe problem formulation. Unfortunately, the quantity of con-straints required is exponential in the number of nodes in theC-Space graph, since the flow through every combination ofnodes in the graph must be inspected to eliminate subtours.For this reason it is impractical to enumerate all of theseconstraints to solve the problem exactly, and an approximatesolution will be found using only (1) and not the exponen-tially many subtour elimination constraints.

The closest relative of this problem which has beenclassified in the operations research literature is known asthe Prize-Collecting Rural Postman Problem (PRPP) [14], anNP-hard problem in which the agent must find the minimum-cost closed walk when there is a prize located on each edgeof the graph that is collected only on the first traversal. Our

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Algorithm 5 CONNECT SUBTOURS(Subtours,qinit ,G)1: q1← qinit2: t1← GET SUBTOUR CONTAINING(q1)3: EligibleSubtours← Subtours\t14: ConnectingPaths← /05: while ConnectingPaths.size()< Subtours.size()−1 do6: {t2,q2}← NEAREST (t1,q1,EligibleSubtours)7: ConnectPath← DIJKST RA(q1,q2,G)8: ConnectingPaths.add(ConnectPath)9: EligibleSubtours← EligibleSubtours\t2

10: q1← q2, t1← t211: end while12: ConnectFinalPath← DIJKST RA(q2,qinit ,G)13: ConnectingPaths.add(ConnectFinalPath)14: return ConnectingPaths

agent also collects a one-time prize from certain edges whenit sees a structure node for the first time, but in our case thecollection of all prizes is a hard constraint and not simply anaddition to the reward. This leaves the inspection problemwith a larger number of constraints than the PRPP.

B. Approximate Formulation for the Optimal Inspection Path

An efficient closed walk for the structure inspection maybe found using (1) only, but to do so the resulting unattachedsubtours must be connected together into a continuousclosed walk. For this purpose, we have designed Algorithm5, CONNECT SUBTOURS(), which iteratively fuses eachsubtour to a neighboring subtour. This algorithm is calledwhen there are two or more subtours returned by the solutionto (1), and it takes as input a list of the subtours, theconfiguration qinit from which the inspection is initialized,and the C-Space graph. Given a starting configuration q1,this algorithm chooses the nearest configuration q2 on a yetunvisited neighboring subtour. Dijkstra’s algorithm is usedto find a feasible path connecting q1 and q2, and q2 is set toq1 for the next iteration. This forces a subtour to connect totwo neighbors through a single point of attachment, whichsimplifies the execution of the closed walk. In this manner,the inspection will begin at qinit, carry out the subtour towhich qinit belongs, and once the agent arrives back at qinit,it will proceed to the next subtour. This pattern repeats,carrying out the next subtour and departing from the sameconfiguration through which it arrived.

IV. INSPECTION PLANNING FOR THE HAUV

A. Platform-Specific Constraints

To implement the inspection planning algorithms for struc-ture inspection by the Bluefin-MIT HAUV, some platform-specific constraints must be formulated. Although the vehicleis capable of motion in surge, sway, heave, yaw, and sonarpitch, we assume that the vehicle always translates using acombination of surge and heave only, since this constraintwill always sweep the largest possible volume with thebathymetry sonar, which is mounted with the beam horizon-tal at zero sonar pitch. Configurations will be sampled in x, y,

Fig. 2. Visual depiction of the sweeping of the bathymetry sonar footprintacross an edge of the graph. The sensor footprint is drawn in green, and thediscrete structure nodes observed by the traversal of the blue graph edgeare plotted in red.

z, and sonar pitch, with the size of the sampling space in x, y,and z dictated by the size of the inspection environment, andsampling in sonar pitch dictated by the allowable limits ofsensor motion (sensor can pitch 90 degrees above or belowthe horizontal plane). The graph edge connecting any twoconfigurations will dictate the vehicle yaw, since translationis constrained to occur in the surge and heave directions only.For every edge in C-Space added to the graph, we constructtwo directed edges, each in which the vehicle starts at a node,adjusts its sonar pitch angle, turns in yaw to face the adjacentnode, then translates in surge and heave to the adjacentnode. Each directed edge sweeps out a different volume ofEuclidean space than its oppositely directed partner. For thepurposes of the minimum-cost flow optimization problem,the cost of each edge of the graph is set to the length ofthe edge in Euclidean space. Pitching the sonar is modeledas a cost-free action since it can occur quickly and withlow energy expenditure. Figure 2 illustrates the observationof some structure mesh nodes during an edge traversal. Weassume the sonar has a 30 degree field of view, a minimumrange of 1m, and a maximum range of 10m.

If we approached this problem from the “art gallery”perspective, sensor observations collected at the nodes ofthe C-Space graph would be the only observations stored.This could be a viable strategy for the HAUV if a noddingmaneuver is performed with the sonar at each graph node,but it discards the valuable sensor information gathered whiletraveling between nodes. Because of the limited field ofview of the bathymetry sonar and the increased expense ofplanning a tour on a larger graph, we have chosen not todiscard this information.

B. Results

The inspection planning algorithms were applied to avariety of structure meshes, which are pictured in Figure3. The meshes were scaled to compare with the largest-sized structures we will inspect with bathymetry sonar duringan autonomous ship hull inspection. In the cases of theactual ship structures (the propeller with nothing attached

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Fig. 3. Four discrete structure models to which the inspection planningalgorithms are applied. The structures are a ship propeller, a sphere, a setof interwoven tubes, and the propeller and rudder of a large ship, whichincludes the complete ship model (in green) for the purposes of obstacleavoidance. Dimensions are in meters.

and the propeller and rudder attached to a hull), the nodesof the structure mesh were increased in density such thatthe maximum distance between structure nodes occupyingthe same triangle was about 0.5m. This spacing betweenthe discrete nodes of the structure model should be chosenbased on the size of objects the inspection is intended todetect. Although we increased the density of nodes, theoriginal, larger triangles were used for collision checkingand ray tracing. For the larger ship mesh, we only requireinspection of the structures possessing severe curvature, andthe remainder of the ship is stored for the purpose of collisiondetection and ray tracing. Figure 4 displays the graphs thatwere constructed to achieve 100% coverage of each structure.For each graph, the minimum-cost closed walk selected asthe inspection path is illustrated in Figure 5.

The only tuning of the algorithm which varied betweenstructures were the boundaries in C-Space from which wesampled the x, y, and z coordinates of each configuration.Otherwise, the same algorithm settings, including a choice often nearest neighbors for the number of connections to makein FIND NEIGHBORS(), and the use of Algorithm 3 for thefinal 5% of the structure, were used for all results depictedhere. IBM’s CPLEX solver was used to solve the integerprogramming problem for the minimum-cost flow. Data oncomputation time and the sizes of the structure meshes andthe C-Space graphs is presented in Table 1. The number ofnodes in each graph scaled approximately with the size ofeach respective structure. Although the propeller mesh wasdensely populated with discrete nodes, the algorithm solved

Fig. 4. The dense graphs constructed for the four example structures ofFigure 3. Each graph is constructed in a 4D configuration space, and isprojected here in 3D Euclidean space. The limits of the space in whichsamples were drawn are identical to the boundaries of the four plots.

for an inspection path quickly and required a relatively sparsegraph to achieve 100% sensor coverage. The other structuresrequired dense graphs by comparison, and the sphere, themost sparsely populated structure model, produced the graphof greatest density due to the ease of connecting each node toits ten closest neighbors without obstruction. The large shipmesh, because of the narrow passages between structures,required significantly more time than the other examples tocomplete graph construction. Most of this time was spentusing Algorithm 3 to search for feasible configurations whichobserved the structure nodes in the narrow cracks in andaround the rudder, which are approximately 0.5m wide inmost locations. Although the tight spaces in this ship modelposed a challenge, as they do for most planning algorithms,the resulting inspection path yielded an intuitively curvedtrajectory to explore the narrow space between the upperpart of the propeller and the hull.

To offer a benchmark for comparison with the results of[8], the prior work most similar to ours, we also apply thealgorithm to a set of randomly chosen cube structures, eachof which is comprised of five nodes and four triangles oneach face, the result of which is plotted in Figure 6. Althoughthe sensor used in our inspections has a rather limited field ofview compared with that of [8], an inspection of comparablepath length and structure is obtained when inspecting thesimpler, dispersed 3D structures assessed in this prior work.

V. CONCLUSION

We have introduced an algorithm for planning inspectionpaths which iteratively constructs a dense graph in the agent’s

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Fig. 5. Inspection paths planned for the four example structures of Figure3. Blue edges represent the subtours required for coverage, and red edgesrepresent the paths selected by Algorithm 5 to connect them.

Fig. 6. An inspection path planned for six randomly planted 1-m cubes,intended for comparison of our algorithm with the result of [8]. Blue edgesrepresent the subtours required for coverage, and red edges represent thepaths selected by Algorithm 5 to connect them.

configuration space until the set of edges observes the entirestructure. A sampling-based approach is favored due to theneed to explore four dimensions comprehensively, and thedesire to avoid doing so exhaustively. To accomodate 3Dstructures of arbitrary complexity we assume a discretemodel of the structure, which can utilize models producedby CAD software or produced by previously collected data.After constructing the graph, we approximate the minimum-cost closed walk along the graph which observes the entirestructure.

Although this algorithm successfully achieves coverageplanning over arbitrary discrete 3D structures and does sousing an information-efficient postman formulation, thereare desired improvements and extensions. We would like to

TABLE IALGORITHM PERFORMANCE FOR THE EXAMPLE STRUCTURES

Propeller Sphere Tubes ShipStructure Nodes 1922 482 1280 3771

Structure Triangles 712 960 2562 2460Graph Nodes 194 785 585 958Graph Edges 806 6014 2786 4078

95% Coverage Time [sec] 32 70 56 187100% Coverage Time [sec] 62 108 142 2792

IP Solution Time [sec] 2 110 97 50Alg. 5 Solution Time [sec] 3 71 24 61

# Steps in Subtours Only 74 295 256 269# Steps in Inspection Path 84 352 305 330

Length of Inspection Path [m] 351 1779 1331 1005

reduce computation time by searching more efficiently forfeasible views of the last few unobserved structure nodes, thestep which is usually most time-consuming. We also wish toquantify and, if possible, optimally manage the division ofcomplexity between the graph construction and path-findingsteps in planning problems of this type, in which an agentmust connect with many thousands of targets rather thanexecute a simple point-to-point path.

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