Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Annual Mechanisms and Robotics Conference

IDETC/MR 2010August 15-18, 2010, Montreal, Quebec, Canada

DETC2010-28905

COMPUTING CONSTRAINED LAPLACIAN NAVIGATION FUNCTION PATHS INCONFIGURATION SPACE

David E. Johnson∗School of Computing

University of UtahSalt Lake City, Utah 84112

Email: dejohnso@cs.utah.edu

Tobias MartinSchool of Computing

University of UtahSalt Lake City, Utah 84112

Email: martin@cs.utah.edu

Elaine CohenSchool of Computing

University of UtahSalt Lake City, Utah 84112Email: cohen@cs.utah.edu

ABSTRACTThis paper demonstrates path-planning for complex geomet-

ric models using harmonic function solutions to Laplace’s equa-tion in the configuration space of the robot. The principal el-ements of the system are an approximate representation of theconfiguration space obstacles, finite element meshing of the freespace, and Laplacian solutions to a path between start and endconfigurations in the free configuration space. Paths found bythis system are smooth and free of local minima. Additionally,the full field solution can be used in novel ways to enforce con-straints on the computed robot path, such as needed for car-likerobots and in the presence of moving obstacles. The system istested on several scenarios, such as a moving, rotating robot anda translating robot with moving obstacles, that demonstrate thegenerality of the approach.

INTRODUCTIONPotential field methods, as introduced by [1], are attractive

robot path planners due to path smoothness and algorithmic sim-plicity. Given an appropriate potential field, the robot need onlylocalize itself in the field and follow the field’s gradient to thegoal. However, simple potential fields are prone to local min-ima, which result in “stuck” robots, even where a path should beavailable. To combat this, navigation functions were proposedby [2], with the desirable property of having a single minimum.Initially, navigation functions had limited applicability, but Con-

∗Corresponding author.

nelly [3,4] and others [5] have extended the original idea by usingmore powerful techniques, such as solving Laplace’s equation onmore complex domains for greater usability.

An important abstraction in path planning is the idea of aconfiguration space C , which is the space of the robot’s degreesof freedom [6]. This idea allows even complex robots to be speci-fied as a point in C . However, obstacles in the robot’s workspacethen map to complex shapes in C , or the set Cobs. As a sim-ple example, a circular robot shrinks to a point in C and all theobstacles in the environment expand by an equal amount whenmapping to Cobs.

Since navigation functions produce a path in the specifieddomain, it is important to consider the shape and orientation ofthe robot as it moves along that path. This paper demonstratespath planning in a fully generalized configuration space, wherethe robot’s configuration is completely specified as a point alonga path. Wang and Chirikjian’s [7] thermal conductivity potentialfield demonstrated the basic applicability of working in a gen-eral configuration space. Other navigation functions have beensolved in the workspace of the robot [3,8], and Pimenta [9] pointsout that most approaches ignore robot orientation and instead justexpand the workspace obstacles by the robot maximum radius.Thus, in general, prior works on navigation functions did notfully characterize the rotational motion of the robot relative tothe obstacles. In contrast, by working in a general configura-tion space, this method can work on different types of robots, inenvironments with moving obstacles, and robots made of seriallinkages instead of just mobile robots.

1 Copyright c© 2010 by ASME

FIGURE 1. HARMONIC FUNCTIONS ARE USED TO SOLVEFOR A LAPLACIAN NAVIGATION FUNCTION IN THE CONFIG-URATION SPACE OF THE ROBOT AND ENVIRONMENT. ADDI-TIONAL CONSTRAINTS, SUCH AS A CAR-LIKE MOTION CON-STRAINT, CAN BE ENFORCED USING THE SOLUTION OVERTHE ENTIRE C-SPACE DOMAIN.

The full Laplacian solution to the navigation problem speci-fies a potential field over the entire problem domain. Often, onlya single path within that domain is desired; however, the addi-tional information from the full field can be used to augment thebasic path tracing of navigation functions. This paper demon-strates using the full solution field to enforce constraints on thecomputed path. In one case, a translating and rotating mobilerobot has a non-holonomic constraint enforced to transform thegeneral path into one suitable for a car-like robot (Figure 1). Inanother case, the full field solution is used to enforce a forwardtime constraint for a translating robot in an environment withtime-parameterized moving obstacles, where time is an explicitdimension in the C-space.

The specific technical contributions of this paper are: 1)Demonstrating a complete pipeline from robot and environmentto Laplacian navigation function in configuration space and backto a path in the robot’s workspace. 2) Showing how the fullLaplacian solution over the domain can be used to enforce con-straints on the robot path, such as non-holonomic and forwardtime constraints.

Our motivation for this project derives from future applica-tion to an omni-directional wheelchair robot, which is charac-terized by a three-dimensional configuration space. Current di-rections in motion planning, which emphasize random samplingin the configuration space, are effective for high-dimensional

problems, but they do not fully specify collision status over thefull domain, and the paths they generate typically are of lowerquality than would be desired for human comfort. We believethe planner developed here applies well to planning for such atask. In one sense, we are trading off efficiency of solution forhigher-dimensional tasks for a more complete solution in lower-dimensions, so that additional constraints on the motion can bespecified and incorporated into the planning solution.

BACKGROUNDWhen potential field planners were first introduced [1], the

potential field was computed as a superposition of potentialfields, one with a minimum at a goal location and others withmaxima at obstacles. The combined field Psum is

Psum = Pgoal +Pobs (1)

The path of the robot would then follow the gradient from aninitial configuration to the goal. This potential field view is iden-tical to computing the path from the line integral of the gradientvector field, which can be thought of as the robot in a physicssimulation pushed and pulled by forces away from and towardthe obstacles and goal, respectively. This superposition of fields(or forces) led to local minima in the field, for example in a “U-shaped” obstacle, or even at the entrance to a narrow passagebetween two obstacles [2].

Navigation functions for path planning were introduced byRimon [2] to remove these local minima. These initial naviga-tion functions were defined on spherical domains with sphericalobstacles, which limited their applicability in practical environ-ments, although some limited mapping into other shapes is pos-sible.

Discrete Harmonic FunctionHarmonic functions describe the behavior of electrical, grav-

itational and fluid potentials. They satisfy the maximum prin-ciple, namely they have no local minima/maxima. This makesthem suitable in Morse analysis [10, 11] and modeling applica-tions [12, 13]. Connolly [4] used them to plan paths for mobilerobots. The approach in [3] uses local numerical solutions onfinite difference grids. [9] recognized that tetrahedral finite ele-ment analysis provides a more general solution domain, as wellas more efficient allocation of computational resources. Oneinteresting aspect was an explicit periodic boundary condition,which elegantly matched a C-space domain when rotational di-mensions are included, however, this was only tested over a do-main with no obstacles.

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Given a domain Ω ⊂ Rn, a harmonic function is a functionu ∈C2(Ω),u : Ω→ R, satisfying Laplace’s equation, that is

∇2u = 0, (2)

where Ω∈Rn and ∇2 = ∑ni

∂ 2

∂x2i

is the Laplace operator. Note

that this paper explores potentials in three-dimensional domains,where ∇2 = ∂ 2

∂x2 + ∂ 2

∂y2 + ∂ 2

∂ z2

APPROACHThe path planning approach taken in this paper is to

1. Approximate the configuration space of the problem by sam-pling the distance field between configurations of the robotand the workspace obstacles. The isosurface of this fieldapproximates the C-space obstacles.

2. Mesh the free space of the configuration space using a con-strained Delaunay tetrahedralization.

3. Use finite element analysis to solve for harmonic functionsolutions to Laplace’s equation over the free space domain.

4. Trace feasible paths between start and end points or regionson the free space domain.

5. Map the path(s) in free space back to the workspace of theproblem to visualize the robot’s path.

By computing the navigation function in the configurationspace of the problem, this approach generalizes to multiple robottypes and scenarios. Note that serial linkages can be used inthe place of a mobile robot in step one by using a different map-ping between configuration parameters and robot geometry in theworkspace, but the rest of the pipeline remains the same. Fur-thermore, by solving the Laplacian over the entire valid domainof the configurations space, several novel improvements are pos-sible, such as forcing the path in C-space to meet additional con-straints.

The steps involved in this approach are detailed in the re-mainder of this section, then the results of testing the approachon several different robot and environmental scenarios will bepresented.

Approximating the Configuration SpaceThe robot and environment are represented in this system

as triangle mesh boundary representations. Given a specified do-main for the robot motion, the configuration space of the problemis sampled at regular grid points, in a style similar to [14]. For ascenario where the robot can translate in the plane and rotate, thiscorresponds to placing the robot at regular intervals of x and y androtating the robot through a sequence of orientations. At eachposition and orientation of the robot, the distance between therobot and environment is computed using PQP’s swept-sphere

FIGURE 2. THE CONFIGURATION SPACE IS SAMPLED ATREGULAR POINTS IN THE DOMAIN TO PRODUCE A DISTANCEFIELD IN THE CONFIGURATION SPACE. THIS IMAGE SHOWSTHE POSITIONS AND ORIENTATIONS OF THE ROBOT IN THEDOMAIN AT THESE SAMPLE POINTS, FOR AN ELLIPTICALDISC ROBOT AND TWO RECTANGULAR OBSTACLES.

hierarchy [15]. Distance measures provide more detailed infor-mation about the configuration space at a given resolution thandoes collision detection. Figure 2 shows an in-progress exampleof computing these samples, albeit on a low resolution grid.

The volume of these samples represents a distance field inthe C-space. The zero isosurface of this field is a linearly in-terpolated discrete approximation to the configuration space ob-stacles. One issue is that penetration depth is not reported bythe distance computation, so colliding objects have a distance ofzero, even when the models are interpenetrated. This aliases theisosurface to go through sample points, possibly into the interiorof the true C-space obstacle.

To counteract this, an isosurface offset is chosen that inflatesthe C-space obstacle boundary the same amount as the worst-case shrinkage possible from the aliasing effect described pre-viously. Since the entire isosurface is expanded, this potentiallyover-inflates other regions of the C-space obstacle. As the dimen-sionality of the problem grows, the possible number of distancechecks needed grows and the impact of the conservative inflationstep increases, as is true for other cell decomposition planners.Again, though, the goal here is to provide a high-quality solu-tion with additional constraint flexibility, rather than a specificplanner for high-dimensional scenarios.

Eventually, the goal is to perform finite element analysis onthe free configuration space. Since C = C f ree +Cobs, C f ree canbe obtained by meshing the C-space domain boundary and sub-tracting the obstacle boundary, yielding a watertight, enclosed

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FIGURE 3. THE SAMPLED DISTANCE FIELD IS ISOSUR-FACED AT A SCALAR FIELD VALUE THAT COUNTERACTSTHE POTENTIAL ALIASING SHRINKAGE FROM LACK OF PEN-ETRATION DEPTH INFORMATION. THE EXTERIOR FACE ISSHOWN IN WIREFRAME TO ALLOW VIEWING OF THE IN-TERIOR OBSTACLE, WHICH IS CLIPPED BY THE DOMAINBOUNDARY.

free space, as seen in Figure 3.

Meshing the Free C-space and Solving for the Naviga-tion Function

Given the domain Ω as computed by the C-space sampling,the interior of the domain is meshed into tetrahedral elementsusing TetGen [16]. We describe a tetrahedral mesh by the tuple(H ,T ,V ,C ) over the domain Ω. H is the set of vertices,v = xv,yv,zv ∈ V ⊂Rn of the tetrahedra in H , and C specifiesthe connectivity of the mesh.

Solving equations for any but the simplest geometries re-quires a numerical approximation. The finite element method(FEM) [17] is used to solve Equation 2 on H . The set V is de-composed into two disjoint sets, VC and VI , representing verticesat which the potential u is known, i.e., the Dirichlet boundary,and vertices for which the solution is sought, respectively. Fur-thermore, the vertices in VC contains start vertices which markthe possible start configurations of the robot and end vertices

FIGURE 4. START AND END CONFIGURATIONS OF THEROBOT ARE SPECIFIED BY ADDING BOUNDARY CONDITIONSTO THE FINITE ELEMENT MESH. NOTE THAT THESE INITIALCONFIGURATION NEED NOT BE SINGLE POINTS, BUT CANALSO BE REGIONS OF INTEREST, SHOWN HERE IN RED ANDGREEN FOR THE START AND END REGIONS, RESPECTIVELY.

which mark the end configurations. This set may be either singlepoints for the start and end, or regions, which gives some flex-ibility in problem specification. For example, Figure 4 shows adomain in configuration space. The vertices in the red area markthe start and the vertices in the green area mark the end. The ver-tices of the white triangles and all interior vertices (not shown)are in the set VI , where the solution to the potential is needed.

Then, in the case of finite elements, solutions are of theform:

u(x,y,z) = ∑vk∈VI

ukθk(x,y,z)+ ∑vi∈VC

uiθi(x,y,z), (3)

where the sums denote weighted degrees of freedom of the un-known vertices, and the Dirichlet boundary condition of the so-lution, respectively. θi(x,y,z) are the linear hat functions, whichevaluate to 1 at vi and to 0 at vi’s adjacent vertices. The weakGalerkin formulation [17] is used to set up a linear systemS~u = ~f , consisting of stiffness matrix S and a right-hand-sidefunction ~f . Since S is positive definite, efficient iterative solutionmethods such as the conjugate gradient method can be used tosolve the linear system [17]. Every point p = (px, py, pz) insideH either lies on the boundary or inside a tetrahedron defined by

4 Copyright c© 2010 by ASME

the vertices vi, and due to the linear basis,

u(p) =4

∑i=1

uiθi(p), (4)

where the ui are the harmonic coefficients of the vertices vi.Lastly, the gradient ∇u over a tetrahedron is the linear combi-nation

∇u(x,y,z) =4

∑i=1

∇θi(x,y,z), (5)

where

∇θi(x,y,z) = (∂θi(x,y,z)

∂x,

∂θi(x,y,z)∂y

,∂θi(x,y,z)

∂ z), (6)

i.e. ∇u is constant over a tetrahedron and changes piecewise con-stantly over the tetrahedral mesh, allowing efficient path tracing.

Figure 5 shows a cut-away interior view of Ω, with bands ofcolor showing the isocontours of the harmonic function solutionto Laplace’s equation. These isocontours curve around the ob-stacles in the C-space, which are holes in the domain. Since thegradient progresses in a direction normal to these isocontours,the robot path also curves away from the obstacles, resulting in acollision-free path in the workplace.

Tracing for Paths Between the Start and GoalGiven a start point in VC, a path is computed by first local-

izing the start configuration inside a tetrahedron in Ω. This tetra-hedron has an associated gradient, ∇u, of the harmonic functionassociated with it. The path follows the gradient vector until itreaches a new tetrahedron, where the new gradient for that tetra-hedron is used. The result is a piecewise linear path through thefree C-space of the problem. Additional smoothing or gradientinterpolation is possible, however, the mesh typically is refinedenough that further smoothing is not necessary. Figure 5 showstraced paths from a start area to goal area through the C-spacegoing around the obstacles in that space. The C-space is shownin a cut-away view for better viewing.

The configuration space path can be mapped back to motionin the workspace and used to control the robot around obstacles.

Constrained Path TracingThe full solution field is largely unused during the tracing of

a single path through C-space, since only the part of the field thatintersects the line integral is evaluated. However, it is importantto realize that the harmonic function provides a field that is di-rected toward the goal in all parts of the C f ree domain. Thus, it

FIGURE 5. THE FREE CONFIGURATION SPACED IS MESHED,GIVEN BOUNDARY CONDITIONS, AND SOLVED FOR HAR-MONIC SOLUTIONS TO LAPLACE’S EQUATION. THIS INTE-RIOR VIEW IN THE LEFT IMAGE SHOWS THE MESH TETRA-HEDRA COLORED BY ISOCONTOURS OF THAT SOLUTION.NOTE THE ISOCONTOURS LEAD FROM THE LOWER RIGHTTO UPPER LEFT OF THE DOMAIN, WHICH WERE THE STARTAND END LOCATIONS DESIRED FOR THE ROBOT. THE RIGHTIMAGE SHOWS SOME REPRESENTATIVE PATHS TRACEDTHROUGH A CUT-AWAY VIEW OF THIS SPACE.

is possible to shift off the current path to a different area of thedomain and still end up at the goal. Arbitrary jumps would leadto physically implausible motions of the robot in the workspace;however, some shifts have realistic consequences.

We use this full-field property to enforce additional con-straints on the robot’s path. For example, car-like motion of arobot requires that its heading is aligned with the tangent to itscurrent path. In configuration space, the robot configuration isa point in a (x,y,θ) domain. As the path is traced through thedomain, the robot’s (∆x,∆y) heading can be compared with theheading specified by θ . If they do not match, then there mustbe tetrahedron with the current (x,y) and the desired θ in the C-space, which can be found by searching above and below the cur-rent point along the θ dimension. If a tetrahedron is not found,then the constrained path must go through an obstacle and a car-like path is not possible.

A similar approach is used in the case of time-parameterizedmoving obstacles. In this case, field flow lines may go aroundthe C-space obstacles in the negative time direction. If a neg-ative time component of the field gradient is detected, then thepath tracing can search in the positive time direction and holding

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FIGURE 6. THIS IMAGE SHOWS A 2D DOMAIN WITH THEVECTOR FIELD FROM THE LAPLACIAN SOLUTION IMPOSED.IMAGINE THAT INSTEAD OF A GENERAL 2D DOMAIN,THE DOMAIN IS INSTEAD A CONFIGURATION SPACE WITHX TRANSLATION AND TIME DIMENSION COMPONENTS. AROBOT FOLLOWING PATH A WOULD REACH A NEGATIVETIME GRADIENT, TRY TO RISE TO A POSITIVE GRADIENT,AND HIT THE OBSTACLE. ROBOTS FOLLOWING PATH B WILLHIT A NEGATIVE TIME GRADIENT, BUT CAN RISE TO AN-OTHER FLOW LINE, SUCH AS C, AND REACH THE GOAL.THUS, THE FULL SOLUTION PROVIDES ADDITIONAL PATHSTO THE GOAL, BUT THE CONSTRAINT MAY IMPOSE SOMEMINIMA IN THE SOLUTION.

(x,y) constant, until a positive time gradient is found (Figure 6).Mapped back to the robot workspace, this is equivalent to hold-ing the robot at the current position until an obstacle moves past,then continuing on. The robot may not be able to reach a validflow line because the additional constraints on the path may in-duce minima in the solution. However, by generalizing the startand end configurations to regions and testing multiple paths, avalid one is likely to be found.

These constrained path techniques are demonstrated in theResults section.

RESULTS

This approach has been tested on two different scenarios.The first is for a robot that translates and rotates in the plane.This robot was placed in a simple maze-like environment. Thesecond scenario allowed just translations of the robot in theplane, however, the obstacles were allowed to move in a time-parameterized, pre-determined motion. These experiments aredetailed in the following sections.

FIGURE 7. ONE OF THE TRACED PATHS FROM THE UPPERLEFT TO LOWER RIGHT IN THE MAZE ENVIRONMENT. THEPATH IN THE CONFIGURATION SPACE INCLUDES THE ORIEN-TATION OF THE ROBOT, ALLOWING IT TO SMOOTHLY BENDAROUND THE WALL CORNERS.

Translating, Rotating, Planar RobotIn the test case for a holonomic, translating and rotating pla-

nar robot, the environment was designed to prevent the robotfrom going around obstacle corners with arbitrary orientations.This creates fairly narrow passages in the corresponding three-dimensional C-space. The robot is a simple elliptical disc shapeand the maze is just two long barriers. Note that the domainboundaries create implicit walls around the maze.

The C-space for this problem was sampled in a 20x20 gridof (x,y) locations in the domain and 15 samples in the orien-tation direction. The rotational dimension of the C-space wasoversampled from −π to 2π . Technically, the rotational dimen-sion should have periodic boundary conditions, which is imple-mentable in this framework [9]. Instead, our explicit unwrappingof the periodicity avoids an occasional unnatural looking motionat the cost of additional sampling time. The C-space obstacleswere created at the zero isocontour, then inflated by 1% of thedomain size to avoid sampling aliasing and to provide a marginof safety to the robot motion around obstacles.

The free space of the domain was meshed for finite elementanalysis, resulting in an FEM mesh with 321400 tetrahedral ele-ments. A collection of paths from the upper-left of the maze tothe lower-right was computed. An example path from that col-lection is visualized in the robot workspace in Figure 7.

Note that the robot path finds one of the limited number oforientations possible for going around the bends of the maze, butin the more open center, the path is underconstrained and the

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FIGURE 8. BY SHIFTING UP OR DOWN IN THE θ DIMENSION,THE TRACED PATH THROUGH THE CONFIGURATION SPACECAN BE CONSTRAINED TO MEET CAR-LIKE ROBOT REQUIRE-MENTS. THIS IMAGE SHOWS THE UNCONSTRAINED PATH INGREEN WITH A CORRESPONDING NON-HOLONOMIC PATH INRED, ALL EMBEDDED IN A CUT-AWAY VIEW OF THE C-SPACE.

robot can choose from a wider number of orientations throughthat section. While this path satisfies the specification of theproblem, additional constraints can be added to produce pathswith different properties.

A Non-Holonomic Car-Like RobotTypically, if potential field planners are used for car-like

robots, the direction vector from the computed gradient of thefield must be combined with complex control laws in the robotto try and maintain the car-like constraints, possibly leading to acomputed path that may be infeasible for a car-like robot to fol-low. In our approach, by maintaining the constraints in the con-figuration space, the computed path is appropriate for the car-likerobot and also maintains a collision-free path.

We use the approach, suggested in the Methods section, ofsearching along the θ dimension of the free space while holding(x,y) constant. This corresponds to steering the robot in the di-rection of travel. A constrained path can be compared to the pathfound without car-like constraints in Figure 8. Note how the cor-rected path has a series of jumps in the θ dimension, correspond-ing to aligning the car-like robot with its direction of travel. Thepath in the physical workspace can be seen in Figure 1.

FIGURE 9. THE ROBOT PATH IN THE PRESENCE OF MOV-ING OBSTACLES CAN BE VISUALIZED WITH TIME USED ASA VERTICAL OFFSET. NOTICE THAT THE ROBOT WAITS FORTHE OBSTACLES TO MOVE PAST BEFORE CROSSING OVER ORMOVES OUT OF THEIR WAY. THE SMALL FRAMES PROVIDESNAPSHOTS OF THE ROBOT’S PROGRESS OVER TIME IN THEWORKSPACE.

Translating Planar Robot with Time-ParameterizedMoving Obstacles

For the next test-case, the robot was reduced to a sphere,since the robot could no longer rotate. The environment used thesame two long barriers as the maze test, but in this case, eachbarrier moved in a line with a sinusoidal motion based on thesimulation clock. In this case, the C-space is still three dimen-sional, with two dimensions for the translational motion, and oneto account for the time-varying position of the obstacles.

The C-space for this test was sampled on a 25x25 grid in thetranslational direction and for 50 time steps in the time dimen-sion. The zero isosurface was inflated by 2% of the domain size,primarily to create a more challenging path planning problem.

The robot was constrained to move from one side of the en-vironment to the other side in a specified time, correspondingto moving from the lower-left of the C-space to the upper right,as seen in the right image of Figure 5. An example path fromthat collection is visualized in the robot workspace in Figure 9,where motion in time is visualized as an offset in the verticaldirection. These paths did not require forward-time constraintsto be enforced, but if a negative time gradient were detected,then a search along the time dimension at the current robot lo-cation would move the path to a flow line which does not violatetime entropy constraints, all while not distorting the motion of

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the robot in the workspace.

Computation TimeThe computation time for these problems includes some pre-

processing time to compute the representation of the configura-tion space obstacles, to mesh the free space, and set up the ini-tial solver. These operations took between 5-15 minutes for thedemonstrated problems on current PCs. Once the preprocess isdone, then new start and end configurations can be specified andsolved in a few minutes, or new paths traced from start to endregions within a few seconds on a single core.

DISCUSSION AND CONCLUSIONThis paper shows a complete pipeline from a robot and envi-

ronmental obstacles, to a navigation function in the configurationspace of the problem, and back to a path in the workspace. Thefull navigation function provides additional information that canbe used to add constraints to the original problem, such as car-like constraints to the motion of a robot in the plane, although ata cost of computational scalability to higher-dimensional scenar-ios. By working in the configuration space, the path through thepotential field fully accounts for the motion of the robot in thepresence of obstacles, making this suitable for a variety of robotsand environments, particularly those needing high-quality paths.

ACKNOWLEDGMENTThis work was supported in part by funding from NSF

(CCF0541402) and NSF (IGERT0654414). All opinions, find-ings, conclusions or recommendations expressed in this docu-ment are those of the author and do not necessarily reflect theviews of the sponsoring agencies.

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