Robot Coverage Path Planning for General Surfaces Using Quadratic Differentials

4.3 MB·

by  Yu-Yao Lin

Robot Coverage Path Planning for General Surfaces Using Quadratic
Differentials
Yu-Yao Lin1, Chien-Chun Ni1, Na Lei2, Xianfeng David Gu1 and Jie Gao1
Abstract—Robot Coverage Path planning (i.e., provide full
coverage of a given domain by one or multiple robots) is a
classical problem in the field of robotics and motion planning. b
Thegoalistoprovidenearlyfullcoveragewhilealsominimize
b
duplicately visited area. In this paper we focus on the scenario a
7 ofpathplanningongeneralsurfacesincludingplanardomains a
1 with complex topology, complex terrain or general surface in
0 3D space. The main idea is to adopt a natural, intrinsic and Fig.1. Thetorusisslicedopenalongthetwogenerators,aandb,ofthe
2 global parametrization of the surface for robot path planning, fundamentalgroupofthetorus.
namely the holomorphic quadratic differentials. Except for a
n small number of zero points (singularities), each point on the
a surface is given a uv-coordinates naturally represented by a to find a path on the cell adjacency graph that visits each
J
complex number. We show that natural, efficient robot paths cell at least once – ideally exactly once (to keep the path

6 canbeobtainedbyusingsuchcoordinatesystems.Themethod short). Finding a path that visits each vertex of a graph
2 is based on intrinsic geometry and thus can be adapted to
exactly once is the well known Hamiltonian path problem,
general surface exploration in 3D.
] which is NP-hard [1]. The adjacency graph may not admit
O I. INTRODUCTION a Hamiltonian path – thus a robot may have to repeatedly
R The Coverage Path Planning (CPP) problem is to deter- visit some points just to get from one cell to the next cell.
. mineapaththatpassesthroughallpointsinagivengeomet- Second, all the algorithms above use the extrinsic coordi-
s
c ric domain. It is a classical problem in robotics and motion nate system, i.e., the Euclidean coordinates representing the
[ planning and is of fundamental value to many applications domain of interest. Such extrinsic coordinate systems, albeit

1 that require a robot or multiple robots to sweep over the beingnaturalchoices,arenotthebesttoencodethecomplex
v target area, such as vacuum cleaning robots, lawn mowers, geometric and topological features introduced by obstacles
9 underwater imaging/scanning robots, window cleaners, and and boundaries. This is in fact the core challenge that the
4 many others. cell decomposition is mean to tackle. When the domain is
5
In general the coverage path planning problem has mul- not flat (e.g., on a terrain or as a general surface in 3D), the
7
0 tiple goals: full coverage (i.e., every point in the domain Ω extrinsic coordinate system and the cell decomposition may
. is covered), no overlapping or repetition (no point is visited lead to unnecessarily many pieces depending on the detailed
1
multipletimes),and/oravarietyofobjectivesonthesimplic- implementation.
0
7 ity or quality of the paths. Satisfying all such requirements
Our Contribution. In this paper we focus on solving this
1 is difficult if not impossible. Therefore priorities are often
: set on these possibly conflicting objectives and the goal is problemusingagenericsolutionthatisapplicabletogeneral
v
surfaces in 3D. The novelty of our method is to abandon
i to obtain a good tradeoff.
X the extrinsic Euclidean coordinates system and adopt the
Thegeometricshapeofthedomaintobecoverediscrucial
r in the design of coverage path planning algorithms. Simple intrinsic coordinate system, i.e., a global parametrization of
a
the domain of interest. To get an idea, consider a standard
shapes such as convex polygons can be covered by simple
torus, one can slice the torus open along the two generators
zigzag motion patterns (lawn mower patterns). Therefore,
of the fundamental group of the torus (See Figure 1 as an
most algoritms for coverage path planning first decompose
example) and the torus can be flattened as a square. Thus,
the target region into ‘simple cells’. The cell decomposition
can be represented by a cell adjacency graph in which each one can represent the points on the torus by a uv coordinate
cell is a vertex and two vertices are connected if they share system,wheretheucoordinaterepresentsthepositionofthe
common boundaries. Within each cell we can use a simple point p along one generator of the fundamental group and
zig-zag pattern and to cover the entire domain we need to v represents the position along the other generator. Both the
geometry of the surface and the topology of the surface are
visit each cell at least once.
inherently encoded in this new coordinate system. Finding a
In all the decomposition methods, there are two general
issues that may affect the final performance. First we need coverage path for the torus under the uv coordinate system
is now trivial – one can simply zig-zag in the uv coordinate
1DepartmentofComputerScience,StonyBrookUniversity,StonyBrook, system which becomes a spiral motion on the torus.
NY,USA{yuylin,chni,jgao,gu}@cs.stonybrook.edu
We introduce the theory and algorithms for computing
2SchoolofSoftware,DalianUniversityofTechnology,Liaoning,China
[email protected] the intrinsic coordinate system using holomorphic quadratic
differentials. Depending on the topology of the surface have even degree has an Euler cycle. In our case, the degree
there are a constant number of zero points (also called of a critical point may not be even. But we can simply
critical points, singular points or singularities) which do double each edge in the graph to create a graph G(cid:48) (which
not have such coordinates. But such singular points are satisfies the requirement) and compute an Euler cycle on G(cid:48)
of zero measure. The coordinate system is naturally repre- –equivalently,eachedgeinGisvisitedpreciselytwice.This
sented by a complex number. One can trace out a curve meanseachcellinthedecompositionisvisitedexactlytwice
by fixing the real/imaginary part of the coordinate, called andwecansimplystretchandshiftthezig-zagpatterninthe
thevertical/horizontaltrajectoryrespectively.Thiscoordinate cell such that the paths followed by the two separate visits
system naturally produces a space decomposition by slicing have minimum overlap.
along critical trajectories (i.e., trajectories that end at zero The main theoretical contribution of this paper lies in the
points). Each component is a simply connected piece with newdiscretealgorithmforcomputingholomorphicquadratic
the complex coordinates as its natural parametrization. This differentials for parametrizing the input domain. In the
decompositioncanalsoberepresentedbyagraphGinwhich literature, a special subset of holomorphic quadratic differ-
the vertices are the critical points and an edge represents entials, named the holomorphic differentials (holomorphic
a cell that touches two singular vertices. This graph and one-forms), have been widely used in computer graphics for
the coordinate system/parametrization are used to generate surface registration and texture mapping [32], [33]. Com-
a coverage path. See Figure 2 for an example. pared to this limited subset, holomorphic quadratic differ-
entials construct a larger family of surface parameterization
with more freedom and flexibility. The algorithm presented
2 here for holomorphic quadratic differentials is new and
9 has never been published before. Mathematically, holomor-
phic quadratic differentials are obtained by multiplying two
6
holomorphic one-forms, and their parameterizations should
1 4 11 satisfy the property of being a curl-free vector field. It is a
5 challenging problem to control the numerical error around
3 8 critical points due to the special local structure. Fortunately
our robot cover path avoids the zero points and all we need
7 10
is to trace out the 3 critical trajectories through each critical
point, which is carefully handled in our algorithm.
(a) ByTrapezoiddecomposition We evaluated the coverage path generated by our al-
gorithms on a variety of different settings including flat
domains with obstacles, non-flat terrains, as well as general
high genus surfaces. Our method is an offline method and
requires the domain to be known in advance.
II. THEORYONHOLOMORPHICQUADRATIC
DIFFERENTIALS
Our solution for the CPP problem is based on a global
surface parameterization, namely the holomorphic quadratic
differentials. Holomorphic quadratic differentials possess a
good property that they inherently induce non-intersecting
trajectories on a surface. This benefit prompts us to develop
(b) ByHolomorphicQuadraticDifferentials
a path planning algorithm based on the trajectories.
Fig.2. Exampleofathreeholesdonutwithtrapezoiddecomposition(2(a)) Holomorphic quadratic differentials form a branch of
and our holomorphic quadratic differentials method(2(b)). In trapezoid
study in complex manifold. In this section, we briefly in-
decomposition, the donut is decomposed into 11 cells, the CPP problem
here is equivalent to find a Hamiltonian path with these cells as vertices, troduce some basics of holomorphic quadratic differentials.
whichisNP-hard.Instead,ourmethodsimplycutsthedonutinto6cells, Then we design our path planning method on general sur-
andtheCPPprobleminoursettingisequivalenttofindingEulercyclewith
faces. For detailed treatments, we refer readers to [34] for
thesecellsasedges,whichcanbeeasilyachievedinpolynomialcomplexity.
Riemann surface theory, [35] for complex analysis, and [36]
for holomorphic quadratic differentials.
Togenerateacoveragepath,weneedtodecidewhatorder
weusetovisitthedecomposedcells.Againweencounterthe
A. Riemann Surfaces
problem of visiting each cell at least once. In our setting we
actually need to visit all the edges (representing the cells) Definition 1: (Manifold). Let M be a topological space.
in G, ideally once and only once. So this is in fact the For each point p ∈ M, there is a neighborhood U and a
α
Euler cycle problem, which is, fortunately, much easier than continuous bijective map φ : U → V from U to an
α α α α
Hamiltonian cycle problem. Any graph in which all vertices open set V ⊂ Rn. (U ,φ ) is called a local chart. If two
α α α
2
neighborhoods U and U intersect, then the transition map This is called the natural coordinate induced by Φ. The
α β
between the chart curves with constant real natural coordinates are called the
vertical trajectories; while the curves with constant imag-
φ =φ φ−1 :φ (U ∩U )→φ (U ∩U )
αβ β α α α β β β α inary natural coordinates are called the horizontal trajec-
tories. A trajectory which ends in zero points is called a
is a continuous bijective map. M is an n dimensional
critical trajectories, otherwise it is a regular trajectory. The
manifold,thesetofalllocalcharts{(U ,φ )}formanatlas.
α α
horizontaltrajectoriesofΦareeitherinfinitespiralsorfinite
Definition 2: (Holomorphic Function). A complex func-
tionf :C→C:x+iy (cid:55)→u(x,y)+iv(x,y)isholomorphic, closedloops.Thismeansthatthetrajectoriesofholomorphic
quadratic differentials are non-intersecting trajectories on a
if it satisfies the following Cauchy-Riemann equation
surface. This property is the key idea of our path planning
∂u ∂v ∂u ∂v algorithm.
= , =− .
∂x ∂y ∂y ∂x Definition 7: (Genus).Agenusgofasurfaceisthelargest
If f is invertible and f−1 is also holomorphic, then f is number of cuttings along non-intersecting simple closed
called a bi-holomorphic function. curves on the surface without disconnecting it.
Definition 3: (Riemann Surface). A Riemann surface is The local structure around a zero point of a holomorphic
a surface with an atlas {(Uα,φα)}, such that all chart quadraticdifferentialisacomplexfunctionz →z23.Forany
transitions φαβ are bi-holomorphic. The atlas is called a holomorphicquadraticdifferentialΦonaclosedsurfacewith
conformal atlas and the local coordinates φα(Uα) are called genus g > 1, there are 4g−4 zero points. For a multiply-
holomorphic coordinates. The maximal conformal atlas is connected surface with n > 2 boundaries, there are 2g−2
called a conformal structure of the surface. zero points of Φ. Zero points are also called critical points
On a Riemann surface, we can define a differential based because they are the endpoints of critical trajectories.
on the conformal structure. Intuitively, a differential can be
C. Surface Decomposition
regarded as a vector field on a surface. The integration on a
differential gives a surface parameterization. The holomor- The path planning technique proposed in this work is
phic differentials and quadratic differentials we introduce applicable to both multiply-connected surfaces(surface with
below are curl and divergence free vector fields. boundaries or obstacles) and general closed surfaces. For
Definition 4: (Holomorphic Differential). Given a Rie- multiply-connected surfaces, we can directly decompose the
mann surface R with a conformal atlas {(U ,φ )}, a holo- surfaces along their critical trajectories. For general closed
α α
morphic differential ζ is a complex differential form defined surfaces, the holomorphic quadratic differentials whose hor-
by a family (U ,z ,ζ ), such that ζ =φ (z )dz , where izontal trajectories are closed loops induce the surface de-
α α α α α α α
φ isaholomorphicfunctiononU ,andifz =φ (z )is composition. The rationale of these properties are described
α α α αβ β
thecoordinatetransformationonUα∩Uβ,thenφα(zα)ddzzαβ = as follows.
φ (z ). Definition 8: (Multiply-Connected Surface). Suppose M
β β
According to the Poincare´-Hopf theorem [37], any vector is a surface of genus zero with multiple boundaries. Then
fieldonasurfacewithnon-zeroEulernumbermusthavethe M is called a multiply-connected surface.
singularities where the vector field vanishes. Such singular-
StrebelDifferentials.Foraclosedsurfacewithgenusg >1,
ities are called zero points. Here we define the zero points
holomorphicquadraticdifferentialsinducethedecomposition
of a holomorphic differential.
for the surface under some conditions. Those holomorphic
Definition 5: (Zero Point). For a point p on a surface R,
quadratic differentials are called Strebel differentials.
if the local representation of a holomorphic differential ζ
Definition 9: (StrebelDifferential[36],[38]).SupposeΦ
s
around p is ζ = φ (z )dz and φ = 0 at p, then p is
α α α α α is a holomorphic quadratic differential on a surface R with
called a zero points of ζ.
genus g > 1. Φ is called a Strebel differential, if all of its
s
regular horizontal trajectories are closed loops.
B. Holomorphic Quadratic Differentials
NoticethatforaStrebeldifferentialΦ onaclosedsurface
S
Definition 6: (Holomorphic Quadratic Differential).
Rwithgenusg >1,alltheregularhorizontaltrajectoriesare
GivenaRiemannsurfaceR.LetΦbeacomplexdifferential
closedloopsasshowninFig.3.Thesetofcriticaltrajectories
form with a conformal atlas {(U ,φ )}, such that on each
α α together with the critical points form the critical graph Γ of
local chart with the local parameter z ,
α Strebel differential Φ . The critical graph Γ decomposes the
s
Φ =φ (z )dz2, surface R into 3g−3 topological cylinders [36].
α α α α
Symmetric Quadratic Differentials. For any given
where φ (z ) is a holomorphic function.
α α multiply-connected surface M with n > 2 boundaries,
1) Zero Points and Trajectories: For a holomorphic
we can find a holomorphic quadratic differential which
quadratic differential Φ on a surface R, any point p ∈ R
decomposes M into 3n − 3 simply-connected surfaces
away from zero has the local coordinate defined as
{d ,d ,...,d }.
1 2 3n−3
(cid:90) p(cid:112) According to the symmetric image property [36], M and
ξ(p):= φα(zα)dzα. (1) its double M¯ form a symmetric surface M˜ = {M ∪M¯}
3
A. Discrete Approximation
The mathematical concepts on smooth surfaces are now
transformed to the numerical procedures on triangular
meshes. A smooth surface is approximated by a piecewise
linear triangle mesh T. The half-edge data structure is
adopted in our implementation. We denote a vertex by v ,
i
a half-edge by [v ,v ], and an oriented triangle face by
i j
[v ,v ,v ].
i j k
Fig.3. TheregularhorizontaltrajectoriesofaStrebeldifferentialareclosed A discrete differential is a function defined on the edge
loopsonthesurface. ω : E → C. The integration of a discrete differential, f :
V →C,givesacomplexnumberorauv-coordinatetoeach
on which their corresponding boundaries are identified. Any vertex.
holomorphic quadratic differential Φ on M is reflected to
B. Algorithm Overview
M¯.Asaresult,AsymmetricsurfaceM˜ iswithasymmetric
holomorphic quadratic differential Φ˜. Because the bound- The following pipeline shows a summary of the main
aries ∂M and ∂M¯ are identified, each horizontal (vertical) procedures of the path planning in this paper. The input is
trajectory γ of M and its symmetric trajectory γ¯ of M¯ are a triangular mesh of a closed surface with genus g > 1,
connected and form a closed loop. or a multiply-connected surface with n > 2 boundaries.
The symmetric surface M˜ is, therefore, a closed surface We first compute the holomorphic differential basis on a
surface, which is then used to compute the holomorphic
with genus g = n. The holomorphic quadratic differential
Φ˜ on M˜ is a Strebel differential, which means the critical quadratic differentials. The holomorphic quadratic differ-
graph decomposes M˜ into 3n − 3 topological cylinders. ential induces a global parameterization, and the resulting
critical trajectories naturally decompose the surface into
Each cylinder c is symmetric along the two curves which
i
3g−3 (3n−3) sub-surfaces. For each sub-surface, we can
are some intervals of ∂M. That is to say, c consists of
i
two symmetric simply-connected domain d and d¯. By compute a number of paths by tracing regular trajectories.
i i
The paths are concatenated together to become a zig-zag
considering {d ,d ,...,d }, we can conclude that the
1 2 3n−3
pathonthesub-surface.Finally,wecombinethesub-surfaces
holomorphic quadratic differential Φ decomposes M into
back to get a continuous path on the whole surface.
3n−3 simply-connected surfaces.
III. ALGORITHM Algorithm 1: Coverage path planning
Input : A triangle mesh T
The core idea of the proposed algorithm is the holomor-
Output: A coverage path planning P of T
phicquadraticdifferentials,whichinducesurfaceparameter-
izationsforgeneralsurfaces.Inbrief,holomorphicquadratic 1 Compute a holomorphic differential basis for T;
differentials inherently induce non-intersecting trajectories 2 Compute a holomorphic quadratic differential Φ for T;
on a surface as shown in Figure 3. This property provides 3 Locate zero points of Φ on T;
us enough freedom on manipulating the trajectories, and 4 Trace the critical graph Γ from zero points;
motivates us to develope our path planning algorithm. 5 T is decomposed along the critical graph Γ and the
sub-surfaces T\Γ={d ,d ,··· ,d } are obtained.
Holomorphic quadratic differentials can be obtained by 1 2 3n−3
For each {d }, generate a path planning P ;
multiplying two holomorphic differentials. III-C briefly i i
lists the computational steps of holomorphic differentials. 6 The path planning of the whole surface is formed by
P ∪P ∪···∪P ∪Γ
Theparameterizationsofholomorphicquadraticdifferentials 1 2 3n−3
shouldsatisfythepropertyofbeingacurl-freevectorfield.It
is challenging to control the numerical error around critical
C. Holomorphic Differentials
points due to the special local structure. As for our robot
cover path, it avoids the zero points and all we need is to The computation of holomorphic differentials is to solve
traceoutthe3criticaltrajectoriesthrougheachcriticalpoint. an elliptic partial differential equation on a triangle mesh
For a topological torus (closed surface with genus one) usingfiniteelementmethod.Thekeystepistousepiecewise
and an annulus, the holomorphic quadratic differentials and linear functions defined on edges to approximate differen-
holomorphic differentials are equivalent. Therefore, by con- tials. Furthermore, the differentials minimize the harmonic
necting each path induced by the trajectories of a holomor- energy, the existence and the uniqueness are guaranteed by
phic differential, a path planning is obtained. The algorithm the Hodge theory [39]. The following algorithm focuses
described in this section focuses on the closed surfaces with on the closed surfaces with genus g > 1. For a multiply-
genus g >1, andthe multiply-connectedsurface with n>2 connected surface with n > 2 boundaries, the algorithm is
boundaries. For a closed surface with boundaries, we can simplified to skip the computation of homology basis [40].
double cover the surface to become a closed surface with Readers can refer to the works by Gu et al. [32], [33] for
genus g >1. Then the algorithm can be directly applied. more details.
4
Algorithm 2: Holomorphic differentials holomorphic quadratic differential decomposes the surface
Input : A closed mesh T with genus g >1 to 3n−3 simply-connected surfaces {d1,d2,...,d3n−3}.
Output: A holomorphic differential basis of T In order to decompose the given surface along the critical
1 Compute the homology group basis {γ1,γ2,··· ,γ2g} graph of a computed holomorphic quadratic differential, we
of T; first locate the zero points on the surface. Then we trace the
2 Compute the dual cohomology group basis critical trajectories from the zero points. Figure 4 illustrates
{ψ ,ψ ,··· ,ψ } of T; the surface decomposition. For a surface with three holes
1 2 2g
3 Compute the harmonic differential basis from the dual (innerboundaries),therearefourcriticalpoints(zeropoints)
cohomology group basis {ψ ,ψ ,··· ,ψ } using heat and six simply-connected domains.
1 2 2g
flow method;
4 For each harmonic differential base ωi, lo√cally rotate by Algorithm 4: Locate zero points
a righ√t angle about the normal to obtain −1∗ωi. Input : A holomorphic quadratic differential Φ on T
ωi+ −1∗ωi forms a holomorphic differential ζi 1 Given a vertex v ∈T, find all vertices connecting to v
sorted counterclock-wisely, denoted as
w ,w ,··· ,w ;
0 1 n−1
In the algorithm below, the holomorphic differential ω + 2 Map wi to the√plane using its natural coordinate
√−1∗ωi is denoted by ζi, where i∈{1,2,··· ,2g}. i ξ(wi):=(cid:82)vwi Φ;
3 The points ξ(w0),ξ(w1),··· ,ξ(wn−1) form a planar
D. Holomorphic Quadratic Differentials polygon and the point ξ(v) is inside this polygon.
Compute the summation of the angles
The holomorphic quadratic differentials on a surface can
be obtained from the products of any two holomorphic n−1
(cid:88)
differentials Φ={ζi·ζj}, i,j ∈{1,2,··· ,2g}. ∠ξ(wi)ξ(v)ξ(wi+1),
i=0
Algorithm 3: Holomorphic quadratic differential where wn =w0. If the summation is 2π, then v is a
regular point; if the summation is no less than 3π, then
Input : A triangle mesh T and two holomorphic
v is a zero point
differentials ζ ,ζ , i,j ∈{1,2,··· ,2g}
i j
Output: A holomorphic quadratic differentials
Φ=ζ ·ζ
i j
1 Compute the products of the holomorphic differentials Algorithm 5: Trace critical trajectories of Φ
ζ ·ζ
i j Input : A holomorphic quadratic differential Φ on T
and a zero point p of Φ
i
1 For pi ∈T, find all faces adjacent to pi sorted
counterclock-wisely, denoted as f ,f ,··· ,f ;
0 1 n−1
p1 d3 p3 2 Fnaotrueraalchcovoerrdtienxatwejξo(wf f)k,:=ma(cid:82)pwjw√j Φto.tThheepclaonmepuustiantgionits
j pi
of natural coordinates is shown in Algorithm 6;
d1 d2 d5 d6 3 The points ξ(w0),ξ(w1),ξ(w2) form a planar triangle,
p2 d p4 where w0 =pi and w0 is mapped to the origin. Let
4 y ,y be the imaginary natural coordinates of
1 2
ξ(w ),ξ(w ) respectively. If y y <0, then the planar
1 2 1 2
triangle, denoted as ∆ , is passed by a critical
ξ
Fig.4. Athree-holedonutwithzeropoints(p1∼p4)andsimply-connected trajectory γ;
surfaces(d1∼d6)decomposedbythecriticaltrajectories(inblue).
4 Compute the natural coordinates starting from ∆ξ. Find
all of the parameterized triangles passed by γ. For a
E. Surface Decomposition g >1 closed mesh, trace γ until hitting a zero point;
For a closed surface with genus g > 1, the surface For a multiply-connected mesh with n>2 boundaries,
decomposition is induced by Strebel differentials. Since trace γ until hitting a boundary;
holomorphic quadratic differentials ζ · ζ form a vector 5 Interpolate the critical trajectory γ, by which the planar
i j
space,andStrebeldifferentialsaretheholomorphicquadratic triangles are passed
differentials with closed horizontal trajectories. Therefore, a
Strebel differential can be computed by the linear combi-
F. Coverage Path
nation of holomorphic quadratic differentials. The surface
is decomposed to 3g − 3 topological cylinders with two The non-intersecting trajectories of holomorphic differen-
boundaries {c ,c ,...,c }. For any multiply-connected tial Φ give the paths for our coverage path planning. Here
1 2 3g−3
surface with n > 2 boundaries, the critical graph of a we take a multiply-connected surface M as an example.
5
Algorithm 6: Compute the natural coordinate of a holo- 9
morphic quadratic differential p1 3 p3
Input : A holomorphic quadr√atic differential Φ on T 12 1 2 8 6 4 5 10
1 Given a face f√∈T, compute Φ. For eac√h edge e of
f, the sign of Φ is decided to satisfy (cid:72) Φ=0 p2 7 p4
√ f
because Φ is a curl free vector field; 11
2 For each vertex v (cid:82)of√f, compute the natural coordinate Fig. 7. An Euler cycle example of the doubled dual graph of Figure 4.
by the integration Φ Thecyclestartsfromp1,andtravelsthroughthearrowededgewithnumber
1∼12,andfinallygoesbacktop1.
Let the outer boundary of M denoted by l , the inner
1 2) Path Interlacement: Euler cycle of the dual graph of
boundaries denoted by {l ,··· ,l }, and the boundaries
2 n a surface implies that every sub-surface is visited twice.
of the decomposed simply-connected surfaces denoted by
By interlacing two zig-zag paths with same density step,
{∂d ,∂d ,··· ,∂d }. Given any density step (cid:15) > 0, if
1 2 3n−3 each sub-surface can be covered nicely. Figure 8 illustrates
l ∩ ∂d (cid:54)= ∅, then we trace a regular trajectory for each
1 i the interlacing paths on the simply-connected domain d in
densitystep(cid:15)alongl ∩∂d .Otherwise,thereexistsaninner 1
1 i Figure4.Therearetwopathstravelingfromonezeropoints
boundary l such that l ∩∂d (cid:54)= ∅, and we trace a regular
i i i totheother,labeledasblueandorange.Whenarobottravels
trajectory for each density step (cid:15) along l ∩∂d . Once the
i i between two zero points, it can choose a color on one way
pathsaregenerated,wecansimplyconnectthepathtogether (as d in Figure 6) and the other color on the other(as d¯ in
1 1
to form a zig-zag path.
Figure 6), hence provide required path density for coverage.
p1 d3 p3
d1 d2 d5 d6
p2 d4 p4
Fig. 5. The dual graph of Figure 4. Here the zero points(p1 ∼ p4) are
treatedasnodes,andthedecomposedsimply-connectedsurfaces(d1∼d6)
thattouchedthesepointsrepresentedgesbetweennodes.
Fig. 8. An Example of paths on the simply-connected domain d1 in
1) Euler Cycle on Surface: Based on our surface decom- Figure4.Therearetwopathstravelingfromonecriticalpointstotheother,
position scheme, we discover that the zero points and the labeledasblueandorange.Whenarobottravelsbetweentwozeropoints,
sub-surfaces can be converted to a dual graph GM. That is, iotncathnecohtohoesre(aas cd¯o1loirnoFnigounreew6)a,yh(eansced1prionvFidigeurreeq6u)ireadndpaththe odtehnesritycofloorr
each zero point is dual to a node and each sub-surface is coverage.
dual to an edge. Moreover, the necessity of visiting every
sub-surface inspires the idea of finding an Euler cycle of Between the adjacent simply-connected domains, a robot
G .Bydoublingeachedge,itisguaranteedtofindanEuler can travel along the critical graph and transfer from one
M
cycle which promises the visiting of every sub-surface. domain to another. By following the path interlacement and
HerewetakethesurfaceshowninFigure4asanexample. the Euler cycle scheme, the coverage path for the whole
Its dual graph is illustrated in Figure 5. Each zero point pi surface is performed. Figure 9 exhibits the coverage path
is dual to a node, and each decomposed simply-connected result for a surface with three inner boundaries.
surface d is dual to an edge. Figure 6 shows the doubled
j
dual graph of Figure 5. For each edge d , the doubled edge
j
d¯ is created. Figure 7 shows an Euler cycle of the doubled
j
dualgraphofFigure4.Onthedualgraph,Eulercyclemakes
the navigation start and end at the same point.
d¯3
p1 d3 p3
Fig.9. Coveragepathforathree-holedonut.
d¯1 d1 d2 d¯2 d¯5 d5 d6 d¯6
p2 d4 p4 IV. EXPERIMENTALRESULTS
d¯4
We evaluate our algorithm on various surfaces, and an-
alyze the influence of different density step on coverage.
Fig.6. ThedoubleddualgraphofFigure4.Wesimplydoubleeachedge
intheoriginaldualgraph(Figure5). We first demonstrate our algorithm on a 2D three holes
donut as in Figure 4. The coverage path result is displayed
6
(a) (cid:15)=8,δ=0.005 (cid:15)=8,δ=0.01
(b) (cid:15)=4,δ=0.005 (cid:15)=4,δ=0.01
Fig.12. Exampleofacoveragepathwithafour-holenon-convexdomain.
(c) (cid:15)=2,δ=0.005 (cid:15)=2,δ=0.01
Fig. 10. The path coverage of three holes donut with different density
step(cid:15)andδ.Theorangeandbluelinesarethecoveragepaths,thesingular
points are labeled as red. Here the covered area is with light blue color.
Noticethat(cid:15)isinverselyproportionaltothepathdensity.
(a) (b)
140 140 Fig. 13. Example of a 3-D terrain with three lakes. The lakes are
Coverage Rate representedbyemptyholes.
120 Overlap Rate 120
%) 100 100
Percentage ( 468000 468000 vaparpiroouaschgersid-abreasceodnaclegronreidthmofsp[r1o2d]u–c[1in5g],aetscm. aMllosntumofbtehresoef
cells in the decomposition, and whether the decomposition
20 20
can be done in the online setting (when the target domain is
0 0
ε= 8 ε= 4 ε= 2 ε= 8 ε= 4 ε= 2 unknown and to be discovered). Other methods include ap-
δ=0.005 δ=0.01
plying spanning tree coverage [16], [17] and neural network
Fig.11. Acomparisonofdensitystep(cid:15)androbotradiusδwithcoverage based coverage [18]–[20].
andoverlaprate.Noticethat(cid:15)isinverselyproportionaltothepathdensity. Coverage path planning for surfaces in 3D is less investi-
gated. Hert et al. [21] considered coverage of a projectively
planar 3D volume, they project the domain in 2D and then
in Figure 10. Here the robot covered area is colored with
take advantages of the 2D planar terrain-covering algorithm
light blue. We fix the robot radius as δ, and the density to solve the problem. Atkar et al. [22] extended the Morse
step (cid:15) represents the step distance on the outer boundary. decomposition to non-planar surfaces but did not consider
Therefore, the bigger (cid:15) brings the sparser coverage paths. obstacles. In [23] Bhattacharya et al. extended their grid-
Notice that even smaller (cid:15) brings better coverage, but it also based algorithm [15] into 3D cased; they first separated the
comeswithapriceofoverlappedcoverage.Acomparisonof
domain into voronoi cells, then handled them by multiple
thetradeoffbetween(cid:15),δ,coveragerateandoverlaprateisas robots.In[24],[25],theauthorsproposedalawnmowertype
Figure 11. As expected, the result shows that the overlap is
ofalgorithmon3Dplanardomain,buttheresultsonlyshow
moreobviousonlargerrobotradiusδanddenserdensitystep terrainswithboundaryandwithoutobstacles.Moreheuristic
(cid:15).Ratherthanthestandard2Ddomain,ouralgorithmisalso algorithmsareadoptedinapplicationscenariosas[24]–[26].
suitable for complex 2D domain and 3D terrain with holes,
Theonemostrelevantwasourearlierworkforgenerating
the result of covering path is demonstrated in Figure 12 and
a space filling curve [27]. However, the focus in [27] was
Figure 13.
to find a curve with progressive density – that is, we want a
path such that the distance from any point to the path to be
V. RELATEDWORK
shrinkingprogressivelywhenthepathgetslonger.Thesame
This problem has been studied extensively and one can as in a followup work [28]. Although quadratic differentials
refer to nice surveys [2], [3] for past work in this area. werealsousedin[28]butboththetheoryandthealgorithms
For 2D domains, most works use a cell decomposition to for generating the curves are totally different from here.
decompose the domain into simple shapes. Popular cell The coverage path problem is also related to various
decomposition includes classical trapezoid decomposition traveling salesman problem (TSP with neighborhoods [29]),
[4], [5] and boustrophedon cellular decomposition [6]–[8], the lawnmower problem (full cover of a region by a path
Morse decomposition [9], [10], slice decomposition [11], with minimum length) [30], and the sweeping path problem
7
(full coverage by a robot arm of fixed geometric degree of [18] C. Luo, S. X. Yang, D. A. Stacey, and J. C. Jofriet, “A solution to
freedom)[31].Sincetheseproblemsaresufficientlydifferent vicinity problem of obstacles in complete coverage path planning,”
inProceedings2002IEEEInternationalConferenceonRoboticsand
we skip the results here.
Automation(Cat.No.02CH37292). IEEE,2002,pp.612–617vol.1.
[19] S. X. Yang and C. Luo, “A Neural Network Approach to Complete
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In this paper, a brand new surface parameterization, holo- Cybernetics, Part B (Cybernetics), vol. 34, no. 1, pp. 718–724, Feb.
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