Motion Planning amidst Fat Obstacles
Motion Planning tussen Vette Obstakels
met een samenvatting in het Nederlands
PROEFSCHRIFT
ter verkrijging van de graad van
do ctor aan de Universiteit Utrecht
op gezag van de Rector Magni cus Prof dr J A van Ginkel
ingevolge het b esluit van het College van Decanen
in het op enbaar te verdedigen
op vrijdag oktob er des middags te uur
do or
Arnoldus Franciscus van der Stapp en
geb oren op oktob er te Eindhoven
Promotor Prof dr M H Overmars
Faculteit Wiskunde en Informatica
CIP GEGEVENS KONINKLIJKE BIBLIOTHEEK DEN HAAG
Stapp en Arnoldus Franciscus van der
Motion planning amidst fat obstacles Arnoldus Franciscus
van der Stapp en Utrecht Universiteit Utrecht
Faculteit Wiskunde Informatica
Pro efschrift Universiteit Utrecht Met index lit opg
Met samenvatting in het Nederlands
ISBN
Trefw geometrie rob otica algoritmen
The research in this thesis was supp orted by the Netherlands Organization for Sci
enti c Research N W O and partially supp orted by the ESPRIT I I BRA Pro ject
ALCOM and the ESPRIT I I I BRA Pro ject PROMotion
Contents
Introduction
The general motion planning problem
Exact motion planning algorithms
Fatness in geometry and thesis outline
Fatness in computational geometry
Fatness
Computing the fatness of an ob ject
Prop erties of scenes of fat ob jects
Fatness implies low density
Arrangements of fat ob ject wrappings
Assembling and disassembling fat ob jects
Fatness de ned with resp ect to other shap es
Range searching and p oint lo cation among fat ob jects
Point lo cation among fat ob jects
Range searching by p oint lo cation
Searching among convex ob jects
Searching among p olytop es
Building the data structure
Summary of results and extensions
The complexity of the free space
The structure of the free space
Results on free space complexities
Fat obstacles and the free space complexity
Existing algorithms and fat obstacles
Boundary vertices retraction
Fatness sensitive cell decomp osition
Complexity of the cell decomp osition
Computing the cell decomp osition
A p olygonal rob ot i
ii CONTENTS
A fatness insensitive cell decomp osition
Boundary cell decomp osition
Towards a general metho d
A paradigm for motion planning amidst fat obstacles
Transforming a base partition into a cell decomp osition
A tailored paradigm for free ying rob ots
E ciently computable base partitions
Arbitrary obstacles in space
Polyhedral obstacles in space
Arbitrary obstacles in space
Similarly sized arbitrary obstacles in space
Planar motion amidst arbitrary obstacles in space
Concluding remarks
References
Bibliography
Index
Acknowledgements
Samenvatting
Curriculum Vitae
Chapter
Introduction
A robot is a machine capable of carrying out a complex series of actions automati
cally the Concise Oxford dictionary Over the past years the use of rob ots has
b ecome common in an increasing number of areas With the wider range of appli
cations comes a growing need for autonomy of the rob ots The earlier generations
of rob ots encountered for example in assembly lines mostly execute prescrib ed
rep eating sequences of uniform actions As such they often e ectively replace
human beings in routine tasks More recent and advanced application domains for
rob ots include op eration in environments that are dangerous or inaccessible to hu
mans Among such domains are space exploration nuclear waste handling and
medical surgery The nature of the rob ot tasks in these environments requires a
high degree of autonomy of the op erational rob ot The series of actions p erformed
by the rob ot tends to b ecome less uniform and the descriptions of the tasks will b e
formulated at a higher level An ultimate goal in the eld of robotics inspired by
this growing need for autonomy is the development of rob ots that accept high level
descriptions of tasks and execute these tasks with as little intervention as p ossible
and ideally without further intervention at all A fundamental task for such an au
tonomous rob ot would b e to move from a current placement to another placement
while avoiding collision with the obstacles on its way The motion planning problem
that is the problem of nding such a collision free path is the sub ject of this thesis
A rob ot is a movable mechanical device op erating in a physical world the rob ot s
workspace Rob ots generally consist of one or more b o dies or links that are in
most practical situations in some way attached to each other These couplings
of the links which are referred to as joints constrain the relative placements and
motions of the attached links Typical joints are the revolute or rotating joint and
the prismatic or sliding joint An articulated rob ot consists of several links that
are all connected by joints If the links of an articulated rob ot are arranged in a
chain and one of the two ends of the chain is xed at some p osition then the rob ot
is an arm The xed end of an arm is referred to as the base of the arm the other
Rob otics is the study of rob ots or the art or science of their design and op eration
CHAPTER INTRODUCTION
end is the tip or hand Rob ots at assembly lines are in general rob ot arms
Typical assembly rob ots have approximately six links The rob ots in the di cult
environments sketched in the previous paragraph are often not xed If except for
p ossible collisions with the obstacles in the workspace or with itself the motion of
the rob ot in the workspace is unconstrained then the rob ot is free ying In this
thesis we will mainly deal with free ying rob ots
The unique characterization of any placement of a rob ot in its workspace involves
a certain minimum number of parameters These parameters are the degrees of
freedom DOF of the rob ot Let us consider the examples of rob ots in Figure
to get a feeling of the various degrees of freedom of rob ots The rob ot arm B moves
L
"
L
#
L
!
q
w
p
L
$
L
O
B B
B
" !
Figure Three examples of rob ots B is a rob ot arm in the plane consisting of
three links B is a free ying articulated rob ot in the plane consisting of two links
!
and B is a free ying rigid rob ot in three dimensional space
"
in a two dimensional workspace and consists of three links L L and L the lower
! "
end of L is xed at the origin O L and L are attached to each other by a revolute
!
joint and L and L are connected by a prismatic joint the overlap of the links L
! " !
and L at the prismatic joint varies b etween and The angle b etween the links
"
L and L and the length of the overlap of L and L uniquely de ne any placement
! ! "
of B so B has two degrees of freedom Any pair w represents
exactly one placement of B As a result the set of p oints in the workspace covered
by B can b e calculated from w provided that the shap es of the individual links
are known The articulated rob ot B with links L and L which are joined by a
! # $
revolute joint moves in a two dimensional workspace Assume for the moment that
the link L is constrained to move at a xed orientation In that case the co ordinates
#
!
x y IR of for example the joint uniquely sp ecify the p oints covered by the
link L The orientation of the link L however is still variable An additional
# $
parameter b eing the angle b etween b oth links L and L completes a
!
unique characterization of the placement of B So the constrained rob ot B has
" "
"
three degrees of freedom Any triple x y IR represents exactly one
placement of the B The triple x y no longer su ces to uniquely sp ecify a
"
placement of B if the link L is allowed to rotate as well Then the rob ot can take
"
in nitely many placements while its joint is placed at x y and the angle b etween its
links L and L equals The addition of an extra parameter giving the
!
angle b etween for example the link L and the p ositive x axis solves the problem
"
"
Any quadruple x y IR sp eci es exactly one placement of this
unconstrained version of B The rob ot B has four degrees of freedom The rob ot
" "
B moving in a three dimensional workspace is a so called rigid rob ot consisting of
#
#
one solid non deformable link A triple x y z IR xes the p osition of some p oint
p B While p is placed at x y z the p oint q can b e chosen to lie anywhere on
#
the sphere with radius jpq j centered at x y z A pair su ces
to identify a p oint on a sphere Even though the quintuple x y z xes b oth
p and q the rob ot B can still b e in in nitely many di erent placements as it is free
#
to rotate around the supp orting line of the segment pq One additional parameter
is enough to mo del this rotational freedom Hence the rob ot B has six
#
#
degrees of freedom Any tuple x y z IR is a
parametric representation of exactly one placement of B We refer to the tuple as
#
a con guration of the rob ot
The motion planning problem is commonly tackled in the space of these para
metric representations of rob ot placements or con guration space for short As we
will see the con guration space formulation transforms the motion planning prob
lem into the problem of nding a continuous curve within a subspace the free space
of the con guration space The free space consists of all placements of the rob ot in
which it intersects no obstacle The continuous curve in the free space corresp onds
to a continuous free motion of the rob ot in the workspace
Motion planning metho ds pro cess the free space for the e cient solution of one
or more path nding queries The metho ds can b e classi ed according to two more
or less orthogonal criteria First al all a metho d is either exact or approximate
Approximate metho ds which originate mainly from the rob otics community are of
ten fast and simple to implement On the other hand they may o ccasionally sp end
a lot of time and storage in nding a path or worse fail to nd a path even if one
exists Exact metho ds which originate mainly from the computational geometry
community are guaranteed to nd a path if one exists The price to pay for this
completeness is generally a considerable increase in computation time A second
sub division classi es the metho ds by the type of technique that is used to nd a
path Latombe distinguishes three di erent motion planning approaches cell
decomposition metho ds roadmap metho ds and potential eld metho ds The next
few paragraphs brie y discuss the essential features of each of the three approaches
Exact and approximate examples of each of the approaches are mentioned if avail able
CHAPTER INTRODUCTION
The cell decomposition approach sub divides a conservative approximation of
the free space FP into a nite number of simple connected subcells such that plan
ning a motion b etween any two placements within a single sub cell is straightforward
and such that uniform crossing rules can b e de ned for the rob ot crossing from
one sub cell into another Each cell de nes a vertex in the connectivity graph CG
Two vertices in CG are connected by an edge if their corresp onding sub cells share
a common b oundary allowing direct crossing of the rob ot Given the connectivity
graph CG the problem of nding a motion b etween the placements Z and Z is
!
reduced to a graph problem nd the sub cells C and C in which Z and Z lie
! !
and determine a path in CG b etween the vertices corresp onding to the sub cells C
and C or rep ort that no such path exists Next the resulting sequence of sub cells
!
and the crossing rules for each pair of subsequent sub cells are used to transform the
sequence into a path for the rob ot B from Z to Z To this end a p oint is chosen
!
on the common b oundary of each pair of consecutive sub cells in the sequence The
p oints corresp ond to unique placements of the rob ot As a result two p oints are
given in every sub cell of the sequence The imp osed simplicity of the sub cells facili
tates the identi cation of a continuous curve b etween the two p oints that is entirely
contained in the sub cell The concatenation of all such curves is a continuous curve
b etween the Z and Z representing a continuous collision free motion for the rob ot
!
b etween the corresp onding placements
Exact cell decomposition metho ds partition the free space into simple sub cells
so that the union of the sub cells equals exactly the free space Examples of exact
cell decomp osition applied to varying instances of the motion planning problem are
found in Section discusses the examples in more
detail Approximate cell decomposition metho ds approximate the
free space by a collection of sub cells with uniform shap es for example rectangloids
The union of the sub cells is a subset of the free space Occasional failure to return
a path is evident from the di erence b etween the free space and the sub cell union
Most approximate metho ds decomp ose the free space in a recursive manner stopping
when a sub cell is entirely free or entirely non free and further re ning when a sub cell
contains b oth types of placements Physical limitations like the amount of storage
that is available require the recursive pro cess to stop at a certain level
The roadmap approach to motion planning aims at capturing the structure
and connectivity of the free space in some one dimensional network of curves the
roadmap The availability of the roadmap reduces the planning problem to de
termining motions b etween the initial and nal rob ot placements Z and Z and
!
two placements on the roadmap and subsequently searching the roadmap for a se
quence of curves connecting these two placements The latter problem is again a
graph searching problem if the network of curves is represented as a graph The
sequence of curves resulting from the graph search corresp onds directly to a contin
uous path for the rob ot in its workspace Nearly all known roadmap algorithms are
exact They share the prop erty that all roadmap curves in a sin
gle connected comp onent of the free space are connected in the roadmap through
a sequence of curves Section reveals some details of certain exact roadmap
metho ds Bro oks presents an approximate roadmap metho d for a translating
and rotating p olygonal rob ot among p olygonal obstacles The basis of the roadmap
is an approximation of the Voronoi diagram on the obstacles in the workspace
Conservative assumptions used in its construction may cause disconnected roadmap
comp onents in a single connected comp onent of the free space leading to p otential
failure to determine a path b etween two placements within a connected comp onent
of the free space The metho d works well if the obstacles are not to o much clut
tered Another approximate roadmap metho d due to Overmars and Svestka
constructs a graph on randomly chosen con gurations in free space Two con g
urations are connected by an edge if a simple collision free motion exists b etween
them
Potential eld metho ds direct the motion of the rob ot through an
arti cial p otential eld set up by the goal placement and the obstacles The goal
con guration pulls the rob ot towards it by generating a strong attractive negative
p otential while the obstacles push the rob ot away through a repulsive p ositive
p otential The search is guided by trusting the intuitive feeling that the direction
of the steep est descent of the p otential is the b est direction towards the goal the
search pro ceeds to a neighboring placement that achieves the maximum decrease of
the p otential The success of the metho d clearly dep ends on adequate choices for
the attractive and recursive p otential functions Unfortunately the search might
get stuck in a lo cal minimum of the p otential Considerable e orts are devoted to
nding ways to deal with these minima One direction of research attempts to sp ec
ify p otential functions that cause no or few lo cal minima Another
approach is to develop techniques to escap e from lo cal minima for example
by random motions Despite the observed complications due to lo cal minima p o
tential eld metho ds are e cient in many practical situations All known variants
of the p otential eld approach are approximate due to fact that steps of a certain
minimum size are taken
Over the past decade the motion planning problem has attracted the interest
of researchers in the eld of computational geometry
The explanation for this interest lies in the geometric
avor of the problem which is not only inherent in its statement but also present
in the space in which the motion planning problem is most conveniently solved
the con guration space The number of degrees of freedom of the rob ot determines
the dimension of this space The con guration space formulation transforms the
motion planning problem into the problem of nding a curve within a subspace the
free space of the con guration space The subspace is the union of sp eci c cells in
an arrangement of hypersurfaces which are de ned by rob ot obstacle contacts
The study of arrangements and arrangement cells is one of the main sub jects
in computational geometry
The research e orts in motion planning in computational geometry are aimed
CHAPTER INTRODUCTION
at the exact solution of the problem so that a path for the rob ot is returned if one
exists It is obvious that the theoretical complexity of nding such a path dep ends
highly on the complexity of the free space The high b ounds on the cumulative
complexity of a collection of cells in an arrangement of hypersurfaces established in
computational geometry demonstrate the p otentially high complexity of the motion
planning problem Even though the hypersurfaces that de ne the arrangement in
the say f dimensional con guration space are not arbitrary as they represent sets
of contact placements they still allow for the construction of worst case arrange
f
ments of roughly at least n complexity where n is the number of obstacles
Exact motion planning metho ds pro cess the appropriate arrangement cells into a
structure capable of providing an exact answer to a path nding query The cumu
lative complexity of the cells is re ected in the size and computation time of the
query structure Since the number of degrees of freedom f of practical rob ots is often
as large as ve or six exact metho ds su er from impractically high computational
costs and are therefore not feasible for practical motion planning problems
On the other hand the worst case arrangements mentioned in the previous para
graph involve arti cial constructions with a rob ot and obstacles with extreme and
often uncommon shap es The complexity of the free space for many real life mo
tion planning problems tends to remain far b elow the theoretical worst case b ounds
Exact motion planning metho ds might b ecome feasible for such realistic problems
provided that their p erformance is p ositively a ected by reductions of the free space
complexity Unfortunately only few of the existing exact motion planning exhibit
such a dep endency The preceding observations show that it is interesting to seek for
mild constraints on the rob ot and the obstacles that lead to a provable low free space
complexity To make the outcome practically useful it is necessary to nd motion
planning metho ds that b ene t from low free space complexities in the sense that
they pro cess the free space in time comparable to its complexity into a path nding
query structure of size comparable to the free space complexity
A b ound on the relative sizes of the rob ot and the obstacles and a certain fatness
of the obstacles are shown to b e su cient to get a free space with a complexity that
is only linear in the number of obstacles Fatness has b een studied in the context of
several problems in computational geometry but so far not in the context of motion
planning Under the sketched circumstances it will b e shown that certain existing
exact motion planning algorithms show a considerable p erformance enhancement
Moreover the realistic assumptions cause the linear complexity free space to have a
structure that allows for a new and simple motion planning paradigm based on the
so called cell decomp osition approach The paradigm basically reduces the planning
problem to a partitioning problem in a lower dimensional subspace Instances of
the paradigm lead to almost linear time in the number of obstacles algorithms
for general planar motion planning and for restricted cases of three dimensional
motion planning namely where the rob ot is con ned to a work oor or where the
sizes of the obstacles di er by at most a constant factor Quadratic and cubic time
algorithms are obtained for three dimensional motion planning among arbitrarily
sized p olyhedral and general obstacles resp ectively The results are indep endent of
the number of degrees of freedom and extend towards any environment with low
obstacle density
We are aware of only few related results on exact motion planning metho ds
with provable e ciency or free space complexity sensitive b ehavior for realistic mo
tion planning problems with low complexity workspaces or free spaces Sifrony
and Sharir present a motion planning algorithm for a line segment in a planar
workspace with p olygonal obstacles The rep orted running time of the algorithm
dep ends nearly exclusively on the number of pairs of obstacle corners that lie less
than the length of the ladder apart This number gives some idea of how cluttered
the obstacles in the workspace are and is furthermore closely related to the com
plexity of the free space Sifrony and Sharir s algorithm is the only algorithm with
a running time that is rep orted to dep end on complexity related variables A few
other algorithms see Chapter have some hidden dep endency on the complexity
of the free space
Schwartz and Sharir consider workspaces with obstacles of so called bounded
local complexity Any imaginary ball with radius r in such a workspace inter
sects no more than a constant number of obstacles The prop erty resembles a
workspace prop erty that follows from the fatness of the obstacles see Chapter
The b ounded lo cal complexity is shown to have implications for the free space com
plexity The authors give directions on how to solve the motion planning problem
in such workspaces
Pignon structures workspaces with p olygonal obstacles for a p olygonal rob ot
to easily detect certain simple and imp ossible path nding queries The author uses
the maximal inscrib ed circle and minimal enclosing circle of the rob ot to de ne the
so called safe and imp ossible spaces which are b oth e ciently computable subspaces
of the workspace The safe space consists of all workspace p ositions that the rob ot
can o ccupy at any orientation without intersecting the obstacles More precisely
the safe space is the collection of center p oints of the enclosing circle in which that
circle do es not intersect any obstacle The imp ossible space consists of all p ositions
in which the rob ot always intersects some obstacle regardless of its orientation
Hence the imp ossible space is the collection of centers of the inscrib ed circle in
which that circle intersects some obstacle The p ossible space is the complement of
the imp ossible space Now two types of simple queries can b e easily detected If
the workspace p ositions of the rob ot in the initial and nal placements b elong to a
single connected comp onent of the safe space then b oth p ositions are connected by
a path for the enclosing circle of the rob ot As a result it su ces to nd a motion
for the circle which is a simpler motion planning problem with two instead of
three degrees of freedom If the p ositions lie in di erent comp onents of the p ossible
space then no motion for the inscrib ed circle of the rob ot exists b etween the query
placements and therefore certainly no motion exists for the rob ot itself b etween
these placements In all other cases the exact solution of the problem requires the
application of an exact metho d to the original problem
CHAPTER INTRODUCTION
Alt et al introduce the tightness of a motion planning problem for a rectangle
among p olygonal obstacles as a measure for its complexity The tightness of a
problem is closely related to the scaling factor for the rectangular rob ot to make
the problem unsolvable if the original problem is solvable or to make the problem
solvable if the original problem is unsolvable The authors present an approximate
motion planning algorithm for the rectangular rob ot with a tightness dependent
running time
The general motion planning problem
This thesis fo cusses on the following version of the general motion planning problem
Given a rob ot B moving amidst a collection of obstacles E and an initial
placement Z and a desired nal placement Z for B nd a continuous
!
motion for B from Z to Z during which the rob ot avoids collision with
!
the obstacles or rep ort that no such motion exists
A single rob ot B moves around in a workspace or physical space W The rob ot s
"
workspace W usually equals the Euclidean space of dimension two or three IR
#
or IR since these are the most interesting cases from a practical p oint of view
Throughout the thesis the robot B is assumed to b e a collection of closed rigid links
attached to each other by joints of constant total complexity A rigid b o dy is a
non deformable compact connected set A closed set incorp orates the set b oundary
as part of the set contrary to an op en set which excludes the set b oundary The
assumption that the rob ot is a collection of rigid b o dies is lib eral as many pap ers
require the rob ot to b e a single rigid b o dy
The motion planning problem is commonly mo deled and solved in the so called
con guration space of the problem The con guration space C is the space of para
metric representations of rob ot placements A con guration Z C is a unique
compact sp eci cation of the p osition of every p oint of the rob ot B at a certain
"
placement in the workspace W Each placement of the rob ot B in its workspace W
corresp onds to exactly one p oint Z in the con guration space C In the sequel the
subtle di erence b etween the con guration Z and the represented rob ot placement
is generally ignored The parameters that are required for a unique sp eci cation of
a rob ot placement x the dimensions of the con guration space These parameters
are referred to as the degrees of freedom of the rob ot The number of degrees of
freedom determines the dimension of the con guration space C At each placement
Z C the rob ot B covers a set of p oints in the workspace W which is denoted by
B Z
For a more concise formulation of the notions of con guration space and con guration in
terms of rigid transformations and relative p ositions of reference frames the reader is referred to Latombe s b o ok on the state of the art in rob ot motion planning
THE GENERAL MOTION PLANNING PROBLEM
Another substantial ingredient of the motion planning problem is the set E of
obstacles in the workspace W Each obstacle E E is a closed connected p ossibly
unbounded subset of W The obstacles are stationary that is they do not move
or change shap e in time Moreover the obstacles of E are assumed to b e known so
that the rob ot B do es not have to explore the workspace W and detect the obstacles
through certain sensing devices The presence of the obstacles in the workspace
causes some placements to b e inaccessible A p oint Z in the con guration space C
can corresp ond either to a placement of the rob ot B in the workspace W in which
it intersects no obstacle B Z E or to a placement of B in W in which
E E
is has non empty intersection with the obstacle set E The rst type of placement
is called a free placement The free space FP is the op en set of all free placements
of the rob ot B hence
FP f Z C j B Z E g
E E
If the placements of the rob ot B are restricted to FP then B is not allowed to move
in contact with the obstacles of E Sometimes however allowing motion in contact
or compliant motion results in more e cient motion planning algorithms A
semi free placement of the rob ot is either a free placement or a placement in which
it touches one or more obstacles but intersects the interior of no obstacle More
formally a semi free placement Z satis es B Z int E where int E
E E
stands for the interior of the closed set E i e E without its b oundary E The
semi free space SFP is the set of all semi free placements of the rob ot B hence
SFP f Z C j B Z int E g
E E
Actually the quoted results solve the motion planning problem in the closure
cl FP of the free space which is formally a subset of SFP Except for some very
sp eci c circumstances see the closure cl FP of the free space equals the
semi free space SFP
The presented problem formulation is extendible in many directions Most of
the generalizations are hardly studied in exact motion planning One extension
is to have non stationary obstacles either moving autonomously see e g or
movable by the rob ot The dynamic b ehavior of the collection of free placements in
the case of autonomously moving obstacles is adequately mo deled by adding a time
axis to the f dimensional con guration space resulting in an f dimensional so
called con guration time space The intersection of this con guration time space
at some time t T shows the free and non free placements at t T The case
of movable obstacles raises additional problematic issues like how to grasp ob jects
Another generalization would b e to allow multiple rob ots An appropriate choice for
the con guration space of such a system of rob ots is the Cartesian pro duct of the
con guration spaces of the individual rob ots see e g Although the results of
this thesis are generalizable towards multiple rob ots we restrict ourselves to a single
rob ot Other extensions are unknown obstacles and non holonomic constraints
CHAPTER INTRODUCTION
Non holonomic constraints are relations b etween the degrees of freedom of the rob ot
The relations imp ose restrictions on the shap e of the collision free rob ot motions
A collision free path or collision free motion or path or motion for short for
a rob ot B from an initial placement Z FP to a nal placement Z FP is a
!
continuous map
FP
with
Z and Z
!
Semi free motion can b e allowed by changing the range of the map into SFP Hence
the problem of motion planning is equal to the problem of nding a continuous
curve b etween two query p oints completely lying inside the free p ortion FP of the
con guration space No quality restrictions with resp ect to length curvature etc
are imp osed up on the rep orted path The e ort that is to b e invested in nding
such a curve obviously highly dep ends on the complexity of the free space FP The
discussion of motion planning algorithms b elow con rms this statement
The complexity of the free space as we will see in Chapter is determined by
the number of multiple contacts of the rob ot B A multiple contact of the rob ot B is
a placement in which it touches more than one obstacle feature that is a basic part
of the obstacle b oundary like a vertex edge or face Besides the collisions of the
rob ot with the obstacles parts of the rob ot can also collide with other rob ot parts
Although these so called self collisions are often ignored in our considerations we
shall return to them at appropriate moments to demonstrate the validity of the
results when self collisions are taken into account Unfortunately the number of
multiple contacts and hence the complexity of the free space can b e very high
If n is the number of obstacle features and f is the constant number of degrees of
freedom of the rob ot that is the dimension of the con guration space and the
number of rob ot features is b ounded by some constant then this complexity can b e
f
n As a generic example consider the rob ot arm in Figure If the lengths
of the links and the distances b etween the obstacles are appropriately chosen then
each of the f links can b e placed against any of the n f obstacles in the vertical row
f
that it cuts through yielding n f combinations of obstacles and therefore leading
f
to n multiple contacts As a consequence the complexity of the free space for
f
the rob ot arm is n Slightly lower worst case free space complexities have b een
obtained for sp eci c free ying rigid rob ots among certain classes of obstacles The
reader is referred to Chapter for an overview of some relevant results These b ounds
f
generally remain close to an order of magnitude i e a factor n b elow the n
b ound Hence even in such b ene cial cases the theoretical worst case b ounds are
high Fortunately in many practical situations the complexity of the free space is
much smaller as arti cially constructed workspaces with e g a very large rob ot and
small obstacles are hardly encountered in real life When extreme shap es and sizes
of the rob ot and the obstacles do not o ccur high free space complexities tend to b e
harder to obtain Consider for example the realistic motion planning environment of
THE GENERAL MOTION PLANNING PROBLEM
n f
f
Figure An f DOF rob ot arm consisting with f links and f revolute joints
f f
with n f fold contacts and hence with free space complexity n
Figure where the spider rob ot and the obstacles have constant complexity and
roughly the same sizes The rob ot has six degrees of freedom two for its p osition
in the workspace and four for each of the legs that are free to rotate around the
central joint While b eing in contact with a certain obstacle the rob ot is unable to
touch more than a constant number of other obstacles on the average Then the
number of multiple contacts can imp ossibly exceed O n Hence the free space for
this rob ot has complexity O n and thus remains far b elow the free space complexity
f
obtained with the construction of Figure The impressive gap b etween the n
B
Figure A DOF rob ot with few multiple contacts and hence with low free
space complexity
construction and the realistic O n example immediately raises the question what
sp eci c prop erties of the rob ot and the obstacles lead to low free space complexities
What natural mild assumptions would for example lead to the relative low obstacle
density of the ab ove example in which the rob ot is unable to touch more than
a constant number of obstacles simultaneously Circumstances that resemble the
relative low obstacle density have b een studied by Schwartz and Sharir who refer
to it as bounded local complexity and by Pignon who calls it sparsity The
case of the DOF rob ot strongly suggests that a b ound on the relative sizes of the
CHAPTER INTRODUCTION
rob ot and the obstacles is necessary to obtain the low obstacle density given that
the obstacles may lie arbitrarily close to each other Comparable rob ot and obstacle
sizes alone however are insu cient to achieve really low free space complexities
A very interesting additional assumption for the obstacles is fatness The fatness
assumption forbids the obstacles to b e long and thin themselves or to have long or
thin parts
Fatness is an interesting phenomenon in computational geometry It has received
quite some attention over the past few years Several pap ers study the surprising
in uence of fatness of the ob jects under consideration on combinatorial and algorith
mic complexities Examples of combinatorial complexity reductions include pap ers
by Alt et al Matou sek et al Efrat Rote and Sharir and Van Kreveld
which all show that the complexity of the union of certain geometric gures is
low if the ob jects are fat Overmars presents an e cient algorithm for p oint
lo cation in sub divisions consisting of fat cells For a discussion of these and some
other results the reader is referred to Chapter For the moment the impact of
fatness is illustrated by a single though very attractive example the complexity
of the union of n triangles in the plane If the triangles are unconstrained then a
quadratic union size can b e obtained by arranging the triangles in a grid like fashion
as shown in Figure Matou sek et al show that the complexity of the union
of n triangles is only O n log log n if the angles of all triangles are at least for
some xed constant
n
n
Figure The union b oundary of n arbitrary triangles can have complexity n
if the triangles are fat see right then the complexity is nearly linear
Chapter prop oses a new notion of fatness that contrary to previous notions
like the fatness for triangles deals with arbitrary shap es in any dimension The
de nition involves a parameter k that gives a qualitative indication of the fatness of
an ob ject The fatness of the obstacles of E according to this de nition along with
a b ound on the relative sizes of the obstacles and the rob ot B and a b ounded com
EXACT MOTION PLANNING ALGORITHMS
plexity assumption for the rob ot and the individual obstacles provide a practical
framework for many real life motion planning problems The free space for all prob
lems that t in this framework is shown to have only linear complexity in Chapter
The linear complexity result op ens the way to devising e cient algorithms for
solving many motion planning problems in realistic environments
Exact motion planning algorithms
Exact algorithms pro cess the free space into a representation that captures all the
necessary details of the structure of the free space to guarantee completeness
The e ciency of such metho ds is usually expressible in terms of the complexity of
the motion planning environment see b elow Judging purely on the worst case
complexities for exact motion planning exact metho ds do not seem to b e practical
alternatives for approximate metho ds in real life situations The constructions that
lead to these complexity b ounds however are hardly encountered in practice In
spired by this observation this thesis shows that under certain realistic assumptions
on fatness and size ratios some exact algorithms do b ecome feasible as their run
ning times are reduced considerably Moreover these realistic assumptions result
in a very b ene cial structure of the free space that allows for an e cient general
paradigm for the exact solution of the motion planning problem
Let us now brie y review the two main classes of exact algorithms cell decom
p osition algorithms and retraction or roadmap algorithms In general the existing
exact motion planning algorithms pro cess the free space into a structure that is
capable of e ciently handling multiple arbitrary path nding queries The run
ning time of an exact motion planning algorithm is actually the time to pro cess the
free space into such a query structure Both exact approaches reduce the motion
planning problem to a graph searching problem Exact cell decomp osition meth
o ds partition the free space FP into a nite number of simple connected subcells
such that planning a motion b etween any two placements within a single sub cell is
straightforward and such that uniform crossing rules can b e de ned for B crossing
from one sub cell into another Applications of the cell decomp osition technique in
clude the famous O n Piano Movers algorithm by Schwartz and Sharir for
planning the motion of a p olygonal rob ot B moving amidst p olygonal obstacles E
in the plane with a total number of n edges This early result has b een improved
!
to O n log n for a ladder line segment rob ot by Leven and Sharir Halp erin
!
!
Overmars and Sharir decide in time O n log n on the existence of a collision
free path for an L shap ed non convex rob ot among p olygonal obstacles Avnaim
Boissonnat and Faverjon apply a variant of the cell decomp osition technique to
a translating and rotating p olygon among p olygonal obstacles Instead of decom
p osing the free space they decomp ose the free space b oundary BFP cl FP n FP
"
in time O n log n The motion obtained with this algorithm is semi free rather
than free except from the rst and last p ortion the rob ot moves in contact with
CHAPTER INTRODUCTION
the obstacles When increasing the con guration space dimension b eyond three
the results deteriorate rapidly Schwartz and Sharir decomp ose the free space
of a rob ot moving amidst p olyhedral obstacles in space The algorithms for a
DOF ladder rob ot and for a DOF p olyhedral rob ot yield connectivity graphs
!
with O n and O n no des sub cells resp ectively and have at least corresp onding
running times where n is the total complexity of the obstacles Ke and O Rourke
give a cell decomp osition algorithm that improves the O n b ound for a lad
"
der in space to O n log n In a di erent pap er in the Piano Movers series
Schwartz and Sharir give a general cell decomp osition algorithm based on algebraic
decomp osition techniques by Collins The running time of the algorithm for a
rob ot with f degrees of freedom and constant complexity amidst obstacles with cu
f !
# $%&"
mulative complexity n is O n which amounts e g to O n for a free ying
rigid rob ot in space Needless to say is that the known results for motion planning
problems with more than three degrees of freedom are far from practical due to their
p erformance Further examples of cell decomp ositions are found in the two other
pap ers in the Piano Movers series
An alternative exact approach to motion planning is the retraction method or
roadmap method The approach recursively retracts the free space FP into a lower
dimensional subspace FP The crucial asp ect of the approach is the retraction
function Im FP FP mapping each placement in FP onto a placement in the
subspace FP A simple collision free motion must exist b etween every p oint Z ! FP
and its mapping Im Z ! FP Provided that such simple motions exist the problem
of planning a motion b etween Z and Z in FP is reduced to the problem of nding a
%
motion b etween their retractions Im Z and Im Z in the lower dimensional space
%
FP Hence motion planning in FP is reduced to lower dimensional motion planning
in FP The ob jective is to obtain after rep eated retractions a one dimensional
network or roadmap N FP There motion planning is reduced to graph searching
if we represent the one dimensional network N as a graph O D unlaingand Yap
and O D unlaing Sharir and Yap present algorithms for planning the motion of
a disc and a ladder based on retractions onto curves in two and three dimensional
#
Voronoi diagrams The algorithms run in time O n log n and O n log n log n
'
resp ectively Leven and Sharir use generalized Voronoi diagrams to extend
the former O n log n result to a translating convex rob ot Sifrony and Sharir
apply a variant of the retraction technique to a translating and rotating ladder
rob ot among p olygonal obstacles They use a retraction that maps placements in
FP onto particular vertices on the b oundary of FP The resulting algorithm runs in
O K log n where K is the number of feature pairs that are less than the length of
the ladder apart Kedem and Sharir present a variant of the retraction approach
for a convex p olygonal rob ot in which they construct a graph on the edges of the
b oundary BFP of the free space The algorithm runs in time O n n log n where
"
i i
log n minf i j log n g where log stands for the logarithm function applied i times in succession
MOTION PLANNING ALGORITHMS
n is a nearly linear function related to Davenport Schinzel sequences
The resulting motions are once more semi free rather than free A general roadmap
based algorithm is due to Canny The algorithm computes a roadmap in the free
f
part of an f dimensional con guration space in roughly O n log n time assuming a
rob ot and obstacles with b ounded complexity The time b ound is close to worst case
optimal
The description of the ideas b ehind b oth exact planning approaches exhibits how
the complexity of the free space in uences the e ciencies of the algorithms As it is
imp ossible to decomp ose the free space into sub cells with a cumulative complexity
that is lower than the free space complexity or to capture the combinatorial structure
of FP in a roadmap with complexity b elow the complexity of FP the complexity of
the free space clearly provides a lower b ound on the complexity and computation
time of any of the motion planning algorithms
A question that immediately comes to mind when considering the linear free
space complexity result is whether it op ens the way to e cient motion planning
algorithms for realistic environments that t in the framework sketched in the pre
vious section The sensitivity to the actual free space complexity of many existing
algorithm is unclear algorithms may e g construct wasteful decomp ositions or
roadmaps in cases of low FP complexity or may construct small decomp ositions
and roadmaps but at relatively high computational cost Two algorithms however
yield a more or less immediate result namely the O K log n b oundary vertices
!
retraction algorithm by Sifrony and Sharir and the O n log n b oundary cell
decomp osition by Avnaim Boissonnat and Faverjon The relative low obstacle
density causes the number K of close corner pairs to b e only O n resulting in a
running time of O n log n for a not to o large ladder among fat obstacles Avnaim
Boissonnat and Faverjon claim that the running time of their algorithm decreases
considerably if the obstacle density is low No other pap ers claim enhanced p erfor
mance of algorithms under certain circumstances
Most of the exact motion planning algorithms discussed in this section have never
"
b een implemented One of the few exceptions is Schwartz and Sharir s O n algo
rithm Ba non discusses an implementation for a ladder rob ot that is rep orted to
p erform surprisingly well contrary to exp ectations based on the theoretical complex
ity analysis This observation may b e due to a hidden sensitivity of the algorithm s
"
running time to the complexity of the free space which is far b elow O n Schwartz
and Sharir do not give any clues in this direction Nevertheless the surprising p er
formance of the exact algorithm motivates a more precise theoretical analysis of its
p erformance under the realistic assumptions sketched in the previous section In
Chapter it is proven that the algorithm by Schwartz and Sharir runs unmo di ed
#
in time O n if the obstacles are fat and the rob ot is not to o large whereas a minor
mo di cation even enhances the e ciency to a running time of O n log n The same
chapter also shows examples of algorithms that do not b ene t from the fatness of the obstacles
CHAPTER INTRODUCTION
Algorithms for e cient motion planning in D workspaces are scarce approaches
in contact space like the algorithms mentioned ab ove by Sifrony and Sharir and
by Avnaim Boissonnat and Faverjon were never shown to generalize to higher
dimensions General approaches to motion planning for example by Canny
and Schwartz and Sharir are computationally exp ensive particularly for the
low free space complexity motion planning problems from the realistic framework
Three dimensional workspaces imply at least three dimensional con guration spaces
with arrangements de ning the free p ortions Naturally the structure of such higher
dimensional arrangements is considerably more complex to understand let alone to
sub divide the free arrangement cells into simple sub cells or catch their structure in
some one dimensional roadmap At this p oint however fatness comes to our help
to provide us with a very b ene cial prop erty of the workspace which in fact also led
to the enhanced p erformance of Schwartz and Sharir s algorithm mentioned in the
previous paragraph the b ounded lo cal complexity of the workspace implied by the
fatness of the ob jects makes it p ossible to partition a subspace of the workspace
W rather than the con guration space into regions R such that the free part of
the con guration space cylinder obtained by lifting R into con guration space has
constant complexity Moreover the b ounded lo cal complexity also establishes the
existence of small partitions into such regions
We formalize and exploit the workspace prop erties outlined in the preceding
paragraph and obtain a paradigm in Chapter for planning the motion of a not to o
large constant complexity rob ot moving amidst constant complexity fat obstacles
The paradigm follows the cell decomp osition approach to motion planning and re
duces the problem of nding a decomp osition of the free space to the problem of
nding a partition of an appropriate lower dimensional subspace of the con guration
space sub ject to some constraints The rob ot s workspace turns out to b e a valid
choice for the subspace under the general circumstances of a free ying rob ot The
size of the free space decomp osition into simple sub cells is determined by the size
of the partition in the lower dimensional subspace of the con guration space The
running time of algorithms based on the paradigm dep ends on the time to nd such
a partition
In Chapter the paradigm is shown to lead to e cient algorithms for many
motion planning problems among constant complexity fat obstacles b oth in IR and
!
IR We brie y review the results Unless stated otherwise the b ounds apply to
free ying rob ots The algorithm for solving the planar problem among arbitrarily
shap ed obstacles in the plane runs in O n log n and outputs an optimal linear
size decomp osition of the free space The same optimal b ounds are obtained for
two practical instances of spatial motion planning The rst case concerns settings
in which the obstacles have roughly the same size that is where the ratios of the
obstacle sizes are b ounded by a constant In the other often encountered case the
obstacles are unconstrained but the motion of the rob ot is con ned to a plane in
the spatial workspace the rob ot s work oor Many examples of such constrained
rob ots can b e found in industrial environments Chapter furthermore rep orts an
FATNESS IN GEOMETRY AND THESIS OUTLINE
O n log n algorithm for motion planning among p olyhedral obstacles in space
The algorithm computes a cell decomp osition of size O n Non p olyhedral obsta
cles require a totally di erent algorithm The simple algorithm presented in Chapter
!
for motion planning amidst arbitrarily shap ed obstacles in space runs in O n
!
time and yields a decomp osition of size O n The results are not restricted to
the sp eci c circumstances of fat obstacles and a b ounded size rob ot but hold in all
workspaces with relative low obstacle densities Note that all b ounds are indep en
dent of the number of degrees of freedom of the rob ot which can easily b e as high
as six or more
Fatness in geometry and thesis outline
The rst chapters of this thesis introduce a new notion of fatness and discuss its role
in a broader geometric context than motion planning From Chapter onward the
emphasis is on the in uence of fatness on di erent asp ects of the motion planning
problem
Chapter introduces a new and general notion of fatness The new notion is
subsequently compared with previous and less general notions The chapter fur
d
thermore rep orts two prop erties of scenes of fat ob jects in IR that are b oth key
to ols in many pro ofs throughout the thesis The rst prop erty applies to scenes of n
non intersecting fat ob jects It is shown that any region of size prop ortional to the
smallest among the ob jects intersects at most a constant number of ob jects in the
scene This prop erty is for obvious reasons rep eatedly referred to as the low ob ject
density prop erty If n non intersecting ob jects are grown then they eventually start
intersecting The second prop erty states that if the growth is again prop ortional
to the smallest ob ject then the arrangement of intersecting b oundaries of the not
necessarily fat grown ob jects has complexity O n Besides its role as a to ol the
latter prop erty has interesting consequences for complexities of union b oundaries
of geometric gures In addition to these results Chapter studies the relation
b etween the lack of fatness of an ob ject E and the lack of fatness of ob jects
E E with E E The main conclusion from the obtained results
" m "im i
is that an ob ject with low fatness cannot b e split into or covered by a constant
number of ob jects with high fatness
In Overmars discusses a data structure for e cient and simple p oint lo cation
in fat sub divisions or sets of disjoint fat ob jects with total complexity n The
d!" d!"
structure supp orts p oint lo cation queries in time O log n and uses O n log n
storage Chapter shows that the data structure can b e used to answer range
queries with arbitrarily shap ed but b ounded size ranges To this end it is proven
that under the condition of fatness of the stored ob jects each b ounded size range
query can b e solved by a constant number of p oint lo cation queries with carefully
d!"
chosen p oints leading to a range query time of O log n It is furthermore shown
that such range queries facilitate the e cient construction of the data structure a
CHAPTER INTRODUCTION
d
problem left op en in in time O n log n log log n
Chapters fo cus on the role of fatness in motion planning A quick glance
of the chapters learns that Chapter concentrates on the combinatorial asp ects of
motion planning Besides giving an overview of combinatorial complexities of var
ious motion planning problems in terms of worst case free space complexities it
formulates the mild assumptions that along with the fatness of the obstacles yield
a linear free space complexity The remaining chapters deal with the algorithmic
asp ects of motion planning In Chapter the impact of fatness on a representa
tive selection of existing planar motion planning algorithms is considered The
algorithms show varying sensitivity to the low free space complexity induced by the
fatness of the obstacles Chapter presents an e cient general paradigm for motion
planning amidst fat obstacles that exploits the sp eci c structure of the free space
of motion planning problems amidst fat obstacles to reduce the problem of nding
a cell decomp osition of the free space to the problem of nding some constrained
partition of a lower dimensional subspace In Chapter the value of the paradigm
is demonstrated as it leads to e cient algorithms for a number of realistic motion
planning problems
Chapter
Fatness in computational
geometry
Many combinatorial and algorithmic worst case complexity b ounds in computational
geometry follow from rather arti cial constructions that are not very likely to o ccur
in practice Often such constructions include extremely small large or thin ob
jects like lines line segments in nitely long simplices and p oints In many cases
the arti cial worst case constructions b ecome imp ossible if the ob jects under con
sideration are not allowed to have extreme shap es but are assumed to have some
fatness prop erty Over the past few years researchers in computational geometry
have not only noted that certain constructions b ecome imp ossible for ob jects with a
certain fatness but more surprisingly also that combinatorial and algorithmic com
plexities of certain problems are provably lower if the ob jects satisfy sp eci c fatness
constraints
A sequence of recent pap ers considers the in uence of fatness in computational
geometry Alt et al Efrat Rote and Sharir and Matou seket al study
the complexity of the union of fat triangles wedges that is regions b ounded by
two half lines emanating from a single p oint and double wedges that is regions
b ounded by two intersecting lines for a xed constant Either one of the regions
is fat if all its internal angles are at least see Figure Note that quadratic
complexity constructions exist for all union sizes if the ob jects in the union are
non fat see for example Figure Alt et al show that the complement of the
union of n fat double wedges consists of O n comp onents In addition they prove
an O n b ound on the b oundary complexity of n homothetic that is scaled and
translated or re ected with resp ect to a vertical line homothetic copies of a single
fat triangle Matou seket al generalize the results by Alt et al The authors
study the union of n fat triangles and prove that its b oundary has complexity
O n log log n The complement of the union consists of O n comp onents If the
triangles have roughly the same size or if they are replaced by fat wedges then
the complexity of the union b oundary b ecomes O n Efrat Rote and Sharir
prove a similar result for the union b oundary of fat wedges with a nearly inverse
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
Figure The triangle wedge and double wedge are fat for some constant
if the indicated angles are at least
quadratic instead of inverse cubic dep endence on the constant Van Kreveld
extends the O n log log n b oundary complexity result to so called wide p olygons
where gives the minimal ratio of the width and length of any corridor
narrow passage in the p olygon Finally Alt et al rep ort an O n b ound for
the complexity of the union b oundary of n translated copies of a b owtie A b owtie
is the rotation gure of a rectangle The linear b ound holds if the rotation angle
do es not exceed arctan b a where a b a b are the lengths of the rectangle s
sides Note that the asp ect ratio of a rectangle that is the ratio of the length of its
sides intuitively provides a go o d qualitative measure of its compactness or fatness
Papers by Katz Overmars and Sharir Overmars De Berg De Gro ot
and Overmars and Agarwal Katz and Sharir rep ort algorithmic conse
quences of the fatness of the ob jects under consideration Katz Overmars and
Sharir present an algorithm for e cient hidden surface removal for scenes of ob
jects with small union size The algorithm computes the visibility map from z
of a set of n fat triangles in space each of which is contained in a plane parallel
to the x y plane in time O n log log n k log n where k is the complexity of
the output Results for comparable scenes of non fat triangles are among others
p
!" #" #
k log n algorithm by Sharir and Overmars and an O n n k an O n
algorithm for any by Agarwal and Sharir Overmars gives a simple
data structure for p oint lo cation in d dimensional sub divisions consisting of fat cells
d ! d !
with query time O log n and storage requirement O n log n For a detailed
discussion of Overmars results and an overview of results on p oint lo cation for non
fat ob jects the reader is referred to Chapter There it is also shown how the p oint
lo cation structure can b e used for range searching and how the data structure is built
e ciently Overmars uses the notion of fatness that is introduced in this chapter
De Berg De Gro ot and Overmars show that an O n size orthogonal sub di
vision of a set of n planar non intersecting fat ob jects exists in which each region
is intersected by a constant number of ob jects The sub division can b e computed
in time O n log n using O n log n storage The result leads to e cient binary
space partitions for scenes of fat ob jects Agarwal Katz and Sharir study depth
orders of non intersecting fat ob jects in space They show that the depth order of
n triangles with fat xy pro jections can b e computed in time O n log n The com
! "#
putation takes O n for any if the triangles are non fat In addition
they prove that the depth order of n convex ob jects with fat xy pro jections and sizes
!
$ %
within a constant ratio from one another is computable in time O n n log n
s
where n is related to the length of so called Davenport Schinzel sequences
s
the parameter s equals the maximum number of intersections of the b oundaries of
any two xy pro jections of the convex ob jects The fatness of the ob ject pro jections
b oils down to a constant ratio b etween the size of the smallest enclosing square and
the largest inscrib ed square of any pro jection Halp erin and Overmars nally
use ideas from the study of fatness to obtain e cient algorithms for manipulating
a molecule mo del of lo osely inter penetrating spheres representing the atoms that
constitute the molecule
The notions of fatness encountered so far mostly apply to a limited set of ob jects
in two dimensional space For our aim a study of the role of fatness in motion
planning we need a more general notion that at least applies to planar and spatial
ob jects The notion of k fatness that is introduced in Section applies to arbitrary
ob jects in any dimension and forbids ob jects to b e long and thin or to have long and
thin parts This type of fatness imp oses a su cient requirement on the obstacles
to obtain a low free space complexity result The parameter k gives a qualitative
indication of the fatness the lower the value of k the fatter the ob ject In the
sequel a fat ob ject is an ob ject that is k fat for a constant k Section furthermore
compares our notion of fatness with alternative notions
The remainder of the chapter is mainly devoted to deducing prop erties that
serve as to ols elsewhere in the thesis Nevertheless some of these results are also
interesting in their own right as they have applications outside motion planning
Section contains the low ob ject density result for scenes of non intersecting
fat ob jects The prop erty plays a crucial role throughout the thesis Within the
same section it forms the basis of a linear complexity result for the arrangement
obtained by growing the fat ob ject b oundaries by an amount prop ortional to the size
of the smallest ob ject Besides its applications in motion planning the latter result
is also interesting in relation to the complexity of the union b oundary of gures in
arbitrary dimension
In Section it is shown that the union of two intersecting fat ob jects is
at most a constant factor less fat than the least fat of its two constituents As a
result it takes at least log k k pieces to partition an ob ject that is not k fat
so thinner than k fat into k fat pieces for any k k The failure to sub divide
a thin ob ject into a constant number of fat ob jects makes it imp ossible to extend
the results in this thesis to thin ob jects by partitioning the ob jects into fat ob jects
as such an approach would increase the asymptotic size of the ob ject set The last
section of this chapter presents a generalization of the notion of fatness and states
the relations b etween the di erent types of fatness that t in the generalization
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
Fatness
Our de nition of fatness in a d dimensional Euclidean space involves d dimensional
closed hyperspherical regions centered at some arbitrary p oint in an ob ject E The
closed hyperspherical region with radius r centered at m will b e denoted by S so
m r
d
S fx IR jd x m r g
m r
the b oundary of S will b e denoted by S so
m r m r
d
S fx IR jd x m r g
m r
Hyp erspherical regions with b oundaries that have non empty intersection with an
ob ject E play a central role in our notion of fatness Therefore the following de
nition is useful
De nition U U
m E E
d
Let E IR be an object The set U is de ned as
E
U U
m E E
m E
where
d
U fS IR j S E g
m E m r m r
So U is the set of all hyperspherical regions with center inside E that do not fully
E
contain E Figure gives two dimensional examples showing two circular regions
S and S that b elong to U and two circular regions S and S that do not b elong
! E " #
to U The region S lies completely inside the ob ject E and is therefore easily seen
E
to b e an element of U The region S is only partly covered by E but since its
E !
center lies inside the ob ject E and its b oundary has non empty intersection with
E the region S is a member of U The circular region S do es not b elong to U
! E " E
b ecause its b oundary has empty intersection with E whereas S is not a member
#
of U b ecause it has its center outside E
E
We de ne fatness in a way such that ob jects are not only compact but also
do not have extremely thin protub erances The de nition of fatness involves some
p ositive number k This number is a measure for the actual fatness of the ob ject
If the value of k is increased then the ob ject is allowed to b e less fat For ob jects
with a b oundary with in nitesimally thin protub erances e g line segments it is
imp ossible to nd such a k so these ob jects can never b e fat
De nition k fat
d
Let E IR be an object and let k be a positive constant The object E is k fat if
S U k volume E S volume S E
FATNESS
S
"
S
!
E
E
m
m
S
S
#
m
E
m
E
Figure Illustration of the de nition of U S S U S U b ecause
E ! E " E
S E S U b ecause its center m E
" # E
Informally an ob ject E is k fat if the part of any hyperspherical region S with a
b oundary that intersects E and its center inside E covered by the ob ject E is at least
a k th of S Hence the relatively emptiest hypersphere among all hyperspheres
centered inside E and with a b oundary intersecting E determines the fatness of E
Figure gives a collection of two dimensional ob jects Below an indication of
the fatness of the these ob jects is given along with an indication of the relatively
emptiest circle The diverse character of the various emptiest circles gives a rst
indication that the fatness of an ob ject may b e hard to compute The ob ject E is
!
not fat due to the in nitely thin part on the upp er edge No nite b ound exists on
the ratio of the area inside the dashed circle and the area of E inside the circle The
!
fatness of a convex ob ject like E is computable from its area and its diameter see
"
Section E is fat The ob ject E may seem quite thin i e not very
" #
fat at rst sight The closeness of the teeth of the comb though makes it hard to
draw relatively empty circles centered inside E E is fat The ob ject E is
# # $
p
fat Like in many other cases the relatively emptiest circle is centered
on the ob ject b oundary and enclosing the ob ject entirely If one of the angles is
chosen smaller then the emptiest sphere is centered at the corresp onding vertex
and passes through the edges incident to the vertex In such a case see e g E %
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
!
!
!
#
!
E E E
! "
! !
!
!"
E E E
# % $
Figure A single non fat ob ject E and ve ob jects E E of varying fatness
! $
The width of a to oth of the comb E equals the distance b etween two successive
#
teeth The width of the narrow bar of the H shap ed ob ject E is times smaller
than the width of b oth wide bars The length of the narrow bar equals the width
of the wide bars The dashed circles are the relatively emptiest circles in U
E
i
i the black dots are the circle centers
the sharp est angle determines the fatness The ob ject E is fat The ob ject E
%
nally is not very fat due to the narrow bar E is fat Here the narrowest
corridor determines the fatness of the ob ject Note that the emptiest circle slightly
p enetrates the wide bars See the next section for some information on computing
the fatness of ob jects
We list a few straightforward prop erties of fat ob jects without pro of as the
validity of each of the prop erties is easily veri ed
d
Prop erty Let E IR be a k fat closed connected object Then
a E is k fat for any k k
b R E is k fat for any rotation matrix R $ S O d
d
c E t is k fat for any translation vector t $ IR
&
d E is k fat for any scaling factor $ IR
FATNESS
The choice for hyperspherical regions in the de nition of fatness is rather ar
bitrary In fact we could have used any compact region with non zero volume
like hypercubic regions regions b ounded by simplices etc Section examines the
relation b etween de nitions of fatness with resp ect to di erent shap es From the
results presented there it is clear that if an ob ject is k fat with resp ect to a given
compact shap e A it is k fat with resp ect to another compact shap e B for some k
that is only a constant multiple of k
The lower b ound on k in the de nition of fatness equals in any dimension d
d
The maximal fatness in dimension d is achieved by the ob ject IR as it covers
of any hypersphere A half space in any dimension d is fat as is covers at least
half of any hypersphere centered inside the half space Determining a lower b ound
on the value of k is a lot more interesting if we restrict ourselves to b ounded ob jects
Then the lower b ound di ers from dimension to dimension There are for example
no b ounded fat ob jects at all there can b e fat ob jects in a two dimensional
workspace but fat ob jects in a three dimensional workspace do not exist Supp ose
we have a k fat ob ject E with diameter The volume of this ob ject is b ounded from
ab ove by the volume of a hypersphere with diameter or radius The diameter
of E is so there is a pair of p oints on the b oundary of E that are a distance apart
let m m E b e these two p oints The hyperspherical region S is an element of
m
U since m S and m E Similarly the hyperspherical region S is an
E m m
element of U Hence the set U contains an element S with radius We know
E E
d d
that volume E ! S volume E and volume S where
d d
is the dimension dep endent multiplier in the volume formulae for hyperspheres
d
d
Combination with De nition E is k fat and S U yields k The
E
d
b oundary value fatness is only obtained for hyperspherical ob jects hyperspherical
ob jects have maximal fatness among the b ounded ob jects
The de nition of k fatness has a lo cal character a certain p ortion of the prox
imity of every p oint in the ob ject must b e covered by the ob ject as well As stated
b efore this lo cality prohibits ob jects with in nitesimally thin protub erances even
if these protub erances are extremely short A huge spherical ob ject with a very
short line segment sticking out of its b oundary will not b e k fat for any value of k
This might contradict with our intuitive idea of fatness An alternative is the more
global type of fatness given in De nition For convenience we will refer to it
!
as thickness Here we only compare the volume of the entire ob ject to the volume
of its minimal volume enclosing hypersphere the volume of the ob ject should b e
at least a certain p ortion of the minimal enclosing hypersphere of the ob ject This
more lib eral de nition allows ob jects with small protub erances If E is an ob ject
then we denote the minimal enclosing hypersphere of E by MES
E
m m
For even dimension m For o dd dimension m
d !m d !m"
See e g Section
!
Thickness is equivalent to our initial notion of fatness as presented in Its shortcomings
with resp ect to the ability to obtain a low free space complexity led to the present de nition of fatness given as De nition
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
De nition k thick
d
Let E IR be an object and let k be a constant The object E is k thick if
k volume E volume MES
E
The de nition of k thickness involves just one hypersphere instead of in nitely many
Note that not necessarily MES U the minimal enclosing hypersphere of an
E E
ob ject can have its center outside the ob ject Again we have the straightforward
prop erty that an ob ject that is k thick is also k thick for k k Spherical ob jects
are thick b ecause the minimal enclosing hyperspheres of such ob jects are the
ob jects themselves
Even though the notion of k thickness seems more natural it is not very useful
for our purp oses b ecause it do es not result in low complexities of the free space due
to the imp ossibility to prove a low ob ject density prop erty for scenes of such ob jects
similar to Theorem which turns out to b e the basis of the low free space
complexity result presented in Chapter We could restrict ourselves to convex
ob jects but as we will see b elow in that case thickness is equivalent to fatness
Therefore we have chosen to use the de nition of fatness stated as De nition
b ecause it also allows for non convex ob jects The prop erty of the set U for
m E
a convex region E given in the next lemma is a useful to ol in the pro of of the
equivalence of thickness and fatness for convex ob jects
d
Lemma Let E IR be a convex object and m E Let S U and
m r m E
S U with r R Now the fol lowing inequality holds
m R m E
volume E S volume E S
m R m r
volume S volume S
m r m R
Pro of We use a p olar co ordinate frame with origin m and angles
d!!
with Each d tuple of angles
d!! d!!
sp eci es a viewing direction from m Since the ob ject E is convex each p oint on
the b oundary of E can b e seen from m Therefore the relation b etween the viewing
direction and the distance to the b oundary of E is a function The same obviously
holds for b oth spheres So there are three functions
E S S
m r
m R
"
d!!
IR f g that give the distance from m to the b oundary of E S
m r
S resp ectively The latter two functions are constant r
m R S d!!
m r
d!!
and R Let f F b e de ned as
S d!!
m R
E d!!
f min
d!!
r
E d!!
F min
d!!
R
The left hand side volume E S volume S is obtained by integrating the
m r m r
pro duct of some determinant function and the function f to some p ower p over
FATNESS
the full angular domain The right hand side volume E S volume S is
m R m R
obtained by integrating the pro duct of the same and the function F to the p ower
j
p over the same domain The determinant is a pro duct of sin terms Since
i
function s range is restricted to Functions f and F
d !
have the same range If we can prove that f F
d ! d !
for all and then b ecause f and F only have
d !
non negative function values the integral containing f will yield a larger value than
the one containing F and hence the inequality involving the volumes will b e proved
Relevant changes in the values of f and F app ear at r and
E d !
R Therefore we consider three di erent ranges for the value
E d !
"
S
m R
S
m r
m
!
E
#
Figure The angular interval is an example of case interval
! ! "
is an example of case and the angular interval is an example of case
" #
of
E d !
If r then
E d !
f r
d ! E d !
R F
E d ! d !
If r R then
E d !
f R F
d ! E d ! d !
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
If R then
E d !
f F
d ! d !
Figure shows a two dimensional example of each of the three cases given ab ove
Combining the three di erent ranges we obtain
f F
d ! d !
for all and
d !
Lemma shows that in each set U the p ortion of a hyperspherical region that is
m E
covered by the ob ject E do es not increase as the radius of the hyperspherical region
increases The ratio is therefore minimal for the region ES U with maximal
m E m E
volume which is the enclosing hypersphere of E centered at m The region ES
m E
is uniquely de ned by following expression
ES U S ES
m E m E S !U m E
m E
A consequence of Lemma is that if k volume E ES volume ES
m E m E
holds then we can conclude that k volume E S volume S for all S U
m E
De ne the set ES of all enclosing hyperspherical regions centered at some p oint in
E
the ob ject
ES fES jm E g
E m E
It is clear that ES U Lemma makes a simpli cation of the condition in
E E
De nition p ossible for convex ob jects Note that for all S ES the obvious
E
equality E S E holds Hence a convex ob ject E is k fat if
S ES k volume E volume S
E
The preceding lemma and considerations provide useful to ols in the pro of of the
equivalence of thickness and fatness for convex ob jects
d
Theorem Let E IR be a convex object Then
" "
E is k fat E is k thick E is l thick E is l fat
" " " "
with k c k and l c l for some constants c and c
Pro of
"
E is k fat E is k thick
Cho ose some hyperspherical region S ES The ob ject E is k fat and
E
ES U so k volume E k volume E S volume S Region
E E
S is some enclosing hyperspherical region of E and MES is de ned as
E
the minimal volume enclosing hyperspherical region of E so obviously
volume MES volume S holds Combining b oth inequalities results
E
" "
in k volume E volume MES proving k thickness of E with k
E
k
FATNESS
E is l thick E is l fat
The convex ob ject E is l thick so l volume E volume MES By
E
Lemma and the convexity of E we know that it su ces to prove that
S ES l volume E volume S for some constant l Let
E
b e the diameter of MES and let b e the diameter of the ob ject E
E
The obstacle E ts inside MES so trivially The diameter of the
E
ob ject E is determined by two p oints m and m on its b oundary The
radius of a hyperspherical region in ES is at most This is the radius
E
of the largest regions ES and ES
m E m E
d
We have volume MES and for all S ES volume S
E d E
d
where is the dimension dep endent multiplication factor men
d d
tioned earlier in this section Combination of all equalities and inequal
ities yields for all S ES
E
d
l volume E
d
volume MES
E
d
d
d
d
volume S
d
proving l fatness of the convex ob ject E with l l
Fatness and thickness are de nitely not equivalent for non convex ob jects as can b e
concluded from the ob ject E in Figure which is not fat but very thick
A consequence of Theorem is that the complexity results that we prove for
convex ob jects that are k fat also hold for convex ob jects that are k thick In the
sequel we will only consider fatness not thickness
The notion of fatness prop osed in this section also relates to most of the other
notions summarized in the introductory part of this thesis for the sp eci c classes
of ob jects to which these other notions apply A fat triangle is also fat
according to our de nition Assume that we are given a fat triangle with a longest
edge e The triangle has minimum area if the other two angles have magnitudes
!
and This minimal area is jej tan Using a result from Chapter on
the fatness of convex ob jects we nd that this triangle is tan fat according
to our fatness de nition Maximum fatness is achieved for equilateral triangles
and there is no fatness if The latter triangle will also b e non fat in
Furthermore it is easily veri ed that a fat wedge is fat and a fat double
wedge is fat Van Kreveld s notion of wideness for p olygons is related for
c gons only where c is some constant It is however rather di cult to determine a
relation b etween the fatness and wideness of a c gon
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
Computing the fatness of an ob ject
Section provides a de nition of k fatness stating when an ob ject is k fat The
fact that an ob ject is k fat however do es not give a clue on how fat the ob ject really
is due to the prop erty that a k fat ob ject is also k fat for all k k The minimum
k for which E is k fat provides a realistic qualitative measure for E s fatness Let
F E b e this minimum so
F E min fk jE is k fatg
We will o ccasionally refer to F E as the fatness of E Substitution of the de nition
of k fatness in the de nition given ab ove yields the following formulation for F E
F E minfk j k volume E S volume S g
S !U
E
volume S
k g minfk j
S !U
E
volume E S
volume S
minfk j maxf jS U g k g
E
volume E S
volume S
maxf jS U g
E
volume E S
The equation shows that the minimum k for which E is k fat is achieved by the
hypersphere S U that maximizes the ratio f S volume S volume E S
E E
Informally this hypersphere S is the relatively emptiest among the hyperspheres
of U Figure shows that the relatively emptiest hyperspheres for the ob jects
E
E E are very di erent
!
d
We will now derive an explicit formula for F E in the case that E IR is an
arbitrary convex shap e with volume V and diameter We were unable to express
the fatness F E of an ob ject E as a function of parameters that are related to the
shap e of E and to characterize the corresp onding relatively emptiest hypersphere s
in U for non convex ob jects
E
The basis for an explicit fatness formula for convex ob jects E lies in Lemma
which after minor manipulations states that the inequality
volume S volume S
m r m R
f S f S
E m r E m R
volume E S volume E S
m R m r
holds for any pair S S U with r R Like in Section we abbreviate the
m r m R E
largest member of U centered at m to E S Notice that E S is the enclosing
E m E m E
hypersphere of E centered at m All enclosing hyperspheres E S centered at some
m E
m E are collected in a set E S A consequence of the ab ove inequality is that for
E
all S U
m r E
f E S f S
E m E E m r
COMPUTING THE FATNESS OF AN OBJECT
Informally the inequality says that the largest of the hyperspheres from U centered
E
at m is the emptiest among all such hyperspheres As a result the relatively emptiest
hypersphere b elongs to the set E S U of enclosing hyperspheres so
E E
maxff S jS U g maxff S jS E S g
E E E E
volume S
jS E S g maxf
E
volume E S
Each hypersphere in E S fully encloses the convex ob ject E hence E S E
E
for all S E S This identity and the assumption volume E V allow for the
E
following reformulation
volume S
maxff S jS U g maxf jS E S g
E E E
V
The constant denominator V of the fraction in the right hand side of the equality
justi es the conclusion that the maximum fraction is obtained when the numerator
volume S is chosen as large as p ossible Therefore the largest hypersphere in E S
E
is the relatively emptiest hypersphere in U Since the diameter of E equals
E
the maximum distance b etween any pair of p oints in E is Let p q E b e such
that the distance from p to q is Then S E S and S E S b ecause
p E q E
q S and p S resp ectively For obvious reasons the set E S contains
p q E
no hyperspheres with radii larger than Combining these considerations with
d d
volume S volume S yields maxfvolume S jS E S g
p q d E d
and thus
d
d
F E max ff S jS U g
E E
V
This leads to the following theorem on the fatness of convex ob jects
d
Theorem Let E IR be a closed convex object with volume V and diameter
d
Then E is V fat
d
The problem of maximizing the ratio f S volume S volume E S for
E
general E or less ambitious for di erent classes of E like p olytop es is very hard
The di culty lies b oth in the shap e and implicit dimension of the domain of
f and in the analytic form of f The continuous domain U of hyperspheres
E E E
of the function f can b e seen as a subset of the d dimensional Cartesian
E
d
pro duct of the d dimensional space of hypersphere centers m m m IR
d
!
and the one dimensional space of radii r IR The complex shap e of the domain
constrained by the two dep endent expressions m E and S E contributes
m r
to the di culty of the problem Another complicating factor is that the analytical
description of f S in terms of m m m and r is not unique throughout
E m r d
the entire domain of hyperspheres due to the changing top ology of the intersection
of S and E m r
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
The problem of computing the fatness for any class of ob jects b eyond convex
shap es remains op en A relaxed version of the problem aimed at nding an upp er
b ound on the fatness that is not more than a constant multiple of the real fatness
that is the maximum ratio volume S volume E S is also largely unsolved
Prop erties of scenes of fat ob jects
In this section we prove two imp ortant results for scenes of fat ob jects The results
form the basis of many pro ofs throughout this thesis In the rst subsection we show
that a scene of non intersecting k fat ob jects satis es a certain low density prop erty
saying that the number of ob jects within a neighborho o d is at most constant The
exact interpretation of neighborho o d is shown to b e dep endent on the sizes of the
ob jects that are involved The low density prop erty resembles the notion of bounded
local complexity introduced by Schwartz and Sharir in
The complexity of the arrangement of the b oundaries of n disjoint constant
complexity ob jects is clearly O n If we expand the ob jects in some way then they
will start intersecting and eventually the asymptotic combinatorial complexity of
the arrangement will increase In the case of general ob jects a drastic increase of
the complexity can o ccur so on after the expansion has started We may exp ect on
the grounds of the low density prop erty of the original disjoint ob jects that such a
sudden increase do es not take place if the ob jects are fat one gets the feeling that an
expansion by some b ounded amount do es not increase the asymptotic complexity of
the arrangement The second subsection provides the circumstances that do indeed
lead to this result The results have immediate consequences for complexities of
union b oundaries like those in and
Fatness implies low density
This subsection discusses a certain low ob ject density prop erty implied by the fatness
of the ob jects under consideration The prop erty has a large impact in the rest of
this thesis as many pro ofs apply it in some form The result can b e paraphrased
in many di erent but essentially similar ways We decide to give two alternative
formulations to save ourselves from rep eatedly deducing either one of the two from
the other one in the future As remarked earlier the result like many others in this
thesis includes a notion of neighborho o d dep ending on the size of the ob jects under
consideration Therefore we shall rst introduce a convenient way to express the
size of an ob ject
Clearly there are many ways to express a b ound on the size of an ob ject The
size that is the radius of the minimal enclosing hypersphere of an ob ject is felt to
b e the most convenient measure for our purp oses The size of the minimal enclosing
hypersphere can b e seen to relate closely to other measures of the size of the fat ob ject like the diameter of the ob ject and b ecause of the fatness also the volume
PROPERTIES OF SCENES OF FAT OBJECTS
The choice for the minimal enclosing hypersphere as a measure of size implies that
if we say that an ob ject X is larger than another ob ject E we implicitly mean to
say that the radius of the minimal enclosing hypersphere MES of X is larger
X
than the radius of the minimal enclosing hypersphere MES of E Similarly the
E
smallest ob ject E means the ob ject with the smallest radius minimal enclosing
hypersp ere MES
E
De nition minimal enclosing hypersphere or mes radius The min
imal enclosing hypersphere radius or mes radius of an object X is the radius of the
minimal enclosing hypersphere of the object X
Theorem states the low ob ject density prop erty for scenes of non intersecting
k fat ob jects
Theorem Let k and c be constants and let E be a set of non
d
intersecting k fat objects in IR with minimal enclosing hypersphere radii at least
Then the number of objects E E intersecting any region R with minimal en
d
closing hypersphere radius c is bounded by the constant k c
Pro of The approach is to identify a region T with b ounded volume such that
each ob ject E intersecting R has a certain minimum volume inside the region T
The combination of the volume of T and the lower b ound on the volume of E T
results in a b ound on the number of ob jects E that intersect R
De ne T MES S the Minkowski di erence of the minimal enclosing
R O
hypersphere MES of R and the hypersphere with radius centered at the origin
R
O The radius of the hypersphere T equals c c which implies that
the volume of T is
d d d
volume T c c
d d
where is the dimension dep endent multiplier for hypersphere volumes
d
Now consider an ob ject E intersecting the region R Let m b e a p oint in the
non empty intersection E R The p oint m lies inside E so it de nitely lies in the
minimal enclosing hypersphere MES as well As a consequence the hypersphere
E
S is completely contained in T MES S In order to b e able to give a lower
m R O
b ound on the volume of E lying in S and hence in T we show that S U
m m E
The ob ject E has non empty intersection with S b ecause m E Moreover
m
the ob ject E cannot lie entirely in the interior of S as this would contradict the
m
assumption that the minimal enclosing hypersphere of E has radius at least So
the b oundary of S is intersected by E and therefore S U From S U
m m E m E
and the containment of S in T it follows that
m
d
volume S k volume E T volume E S
m d m k
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
d d d
The combination of volume T c and volume E T k
d d
d
for any E intersecting R results in an upp er b ound of k c on the number of
ob jects E intersecting R
Informally the theorem states that the number of k fat ob jects intersecting a region
that is not to o large compared to the ob jects is constant The weakness of the notion
of thickness as de ned in the previous section lies in the imp ossibility to deduce
a similar prop erty for scenes of such ob jects This imp ossibility is illustrated by
the two dimensional example of Figure where an extremely small rectangular
region is intersected by n very thick namely thick ob jects In a motion planning
context this would imply that even an extremely small rob ot is able to touch many
obstacles simultaneously which p otentially leads to a high complexity free space
R
Figure The small rectangular region R intersects n thick ob jects
The following alternative formulation of the low density prop erty in terms of
distances can b e given It b ounds the number of larger k fat ob jects that can lie
close to a given k fat ob ject
Corollary Let k and c be constants and let E be a set of non
d
intersecting k fat objects in IR Let E E be an object with minimal enclosing
!
hypersphere radius Then the number of object E E with larger minimal en
closing hypersphere radii within a distance c from E is bounded by the constant
d
k c
!
Pro of Any ob ject E within a distance c from E must also lie within the same
distance c from the minimal enclosing hypersphere MES of E which has radius
E
!
Necessarily such an ob ject E must then intersect the region b ounded by the hy
p ersphere concentric to MES but at a distance c from MES hence with radius
E E
c So application of Theorem with R chosen to b e the region b ounded
by the concentric hypersphere with mes radius c yields the claimed result
PROPERTIES OF SCENES OF FAT OBJECTS
Arrangements of fat ob ject wrappings
This subsection studies the complexity of the arrangement of b oundaries of ex
panded fat ob jects Clearly the arrangement of the b oundaries of n non intersecting
constant complexity ob jects has O n complexity Let us see what happ ens if the
ob jects are expanded While expanding the fat ob jects each of the b oundaries will
eventually start intersecting other b oundaries Intuitively the rst b oundaries that
are to b e intersected b elong to neighboring ob jects As there is only a constant num
b er of ob jects closeby the contribution of each b oundary to the complexity of the
arrangement of b oundaries do es again intuitively not increase asymptotically as it
starts intersecting the b oundaries of these ob jects Below these informal ideas are
made concrete by giving accurate b ounds on the expansion of the k fat ob jects such
that the combinatorial complexity of the arrangement of the b oundaries of these in
tersecting expansions equals O n The so called wrappings that are introduced
rst provide a convenient means of expressing the expansion of an ob ject
Su ciently tight wrappings of fat ob jects play a crucial role in providing the
justi cation that the paradigm for motion planning amidst fat obstacles presented
in Chapter indeed works Besides that the wrappings also help in nding e cient
instances of the paradigm for sp eci c classes of motion planning problems More
over the theorem on wrappings that we prove b elow is interesting in its own right
as it implies nice complexity b ounds for certain arrangements and for the b oundary
of the union of sp eci c families of shap es
De nition wrapping
d
Let E IR and let IR Any object E satisfying d p E for al l p
is an wrapping of E
An wrapping of an ob ject E is an enclosing shap e of E with the prop erty that
the distance from the wrapping to E never exceeds
Theorem states the circumstances that lead to a linear complexity arrange
ment of expanded fat ob ject b oundaries An obvious way to express a b ound on the
expansion of an ob ject E is to state that the expanded ob ject is some wrapping
of the ob ject E itself for some b ounded p ositive Note that the ob ject expansions
need not necessarily b e fat
Theorem Let k and c be constants and let E be a set of n non
d
intersecting k fat objects in IR with minimal enclosing hypersphere radii at least
Assume that a constant complexity c wrapping E is given for every object
E E Then
a the complexity of the arrangement A of al l wrapping boundaries E is
O n
d
b every point p IR lies inside at most O wrappings E
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
Pro of To prove the a part let us assume that the ob jects in E are ordered
by increasing size E E and that are the corresp onding minimal
n n
enclosing hypersphere radii so We intend to count for each
n
ob ject E the subspaces of dimensions to d that are de ned by the intersection
i
of E and wrapping b oundaries E with j i A c wrapping b oundary
i j
E can only b e intersected by c wrapping b oundaries E i j if
i j
the distance from E to E do es not exceed c c See Figure for a
i j i
D example of intersecting wrappings Application of Corollary yields that
E
j
x E
k
E
i
c
i
i
Figure The b old lines are the b oundaries of c wrappings and b ecause
also c wrappings of the ob jects E E and E The set of ob jects
i i i j k
with wrapping b oundaries that intersect the wrapping b oundary of E is a subset
i
of the ob ject within a distance c c from E Although E lies a distance
i i k
x c from E E s wrapping b oundary do es not intersect E s wrapping
i i k i
b oundary
there can only b e a constant number of such E s within a distance c from E
j i i
so there is at most a constant number of wrapping b oundaries E j i that
j
intersect E By the additional assumption that all wrappings have constant
i
complexity there is only a constant number of subspaces of dimension b etween
and d de ned by the intersection of E and wrapping b oundaries E
i j
j i Adding the contributions of all wrappings amounts to a total of O n
subspaces of dimensions to d in the arrangement A The linear b ounds
on the number of these subspaces imply the same b ound of O n on the number
of d faces in A making the total combinatorial complexity of the arrangement
O n
The b part follows immediately from the pro of of the a part Let E b e the i
ASSEMBLING AND DISASSEMBLING FAT OBJECTS
d
smallest ob ject for which the p oint p IR lies inside the wrapping E Since
i
there is only a constant number of wrappings of larger ob jects intersecting E s wrap
i
ping the p oint p can b e in no more than a constant number of additional wrappings
Besides applications in motion planning that b ecome clear later in this thesis
Theorem has interesting implications for complexities of union b oundaries of
certain geometric gures The relation b etween the complexity of an arrangement
of wrapping b oundaries and the complexity of the b oundary of the union of the
wrappings b ecomes clear if one realizes that the faces of the union b oundary form
a subset of the faces of the arrangement of wrapping b oundaries So under the
circumstances sketched in Theorem the b oundary complexity of ! E
E E
is O n An alternative more or less inverse informal formulation of the result
is the following The b oundary complexity of the union of intersecting constant
complexity ob jects is linear in the number of ob jects if k fat sub ob jects can b e
identi ed in all ob jects such that the sub ob jects are mutually non intersecting and
not more than some b ounded amount smaller than the original ob jects The b ounded
amount must b e prop ortional to the size of the smallest ob ject As an example the
ideas are applicable to the molecule mo del in the pap er by Halp erin and Overmars
The atoms that constitute a molecule are assumed to satisfy the hard sphere
mo del The hard sphere mo del describ es atoms by spheres and forbids any sphere
center to p enetrate another sphere to o far This prop erty makes it p ossible to regard
the atoms as wrappings of certain non intersecting smaller spheres which are only
a b ounded amount smaller than the original atoms The construction provides an
alternative pro of for the linear in the number of atoms descriptional complexity
of the molecule surface
Assembling and disassembling fat ob jects
The ob jective in this section is to show that it takes more than a constant number
of cuts to partition a thin ob ject into fat parts In fact we show that the minimum
number of parts that is needed to cut up a thin ob ject into fat sub ob jects is at
least logarithmically dep endent on a measure of the lack of fatness of the ob ject
The main impact of this result is that it is imp ossible to extend the results in the
remainder of this thesis for fat ob jects to thin ob jects by simply partitioning the
thin ob ject into fat ob jects without increasing the total number of ob jects
To get to the ab ove result we study the implications for fatness of splitting an
ob ject into two sub ob jects We subsequently observe that a very fat ob ject can b e
split into two extremely thin ob jects that a thin ob ject can b e split into two ob ject
of which one can b e very fat and nally that at least one of the parts resulting
from splitting a thin ob ject cannot b e more than a constant factor fatter or less
thin than the original ob ject The latter result forms the basis of the main result
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
mentioned in the rst paragraph
Before we move on to examine the e ect of splitting we must rst nd an
appropriate way of expressing the b ound on the lack of fatness or thinness of an
ob ject as the observations include thin ob jects as well as fat ob jects A problem
lies in the fact that the remark that an ob ject is k fat do es not say exactly how fat
the ob ject is but only that the ob ject is not less fat or thinner than k fat On
the opp osite we would now also like to have a means of expressing that an ob ject
is not less thin or fatter than a certain amount Similarly saying that an ob ject
is fat do es not necessarily mean that it is very thin b ecause the ob ject
might still b e fat as well Fortunately the negation do es supply a b ound on the
extent to which an ob ject is fat The fact that an ob ject is not k fat tells us that it
is de nitely not less thin or fatter than k fat Hence the negation of fatness supplies
an appropriate means of expressing a certain guaranteed amount of thinness or lack
of fatness
A simple example shows that it is p ossible to partition a very fat ob ject into two
unboundedly thin sub ob jects Take the fat circle E of Figure The circle is
E
E
E
E
!
E
E
!
Figure The left fat circle E is split into two arbitrarily thin ob jects E and
E The right relatively thin ob ject E is split into a fat ob ject E and another
!
ob ject E
!
split equivalent parts each one b eing half a circle with a thin needle sticking into the
opp osite half The needle can b e made as thin as one likes resulting in extremely
thin sub ob jects of the circle S The example straightforwardly generalizes to higher
dimensions Obviously it is also p ossible to partition a very fat ob jects into two
parts of which exactly one is unboundedly thin
A second observation is that a very thin ob ject can b e split into two parts of which
one is extremely fat Consider the second example in Figure of a rectangle E in
D with very large asp ect ratio i e the ratio of its side lengths The partitioning
into two parts of which one is extremely fat can b e obtained by simply cutting an
arbitrarily small fat circle E out of the rectangle Again the D example is
straightforwardly generalizable
ASSEMBLING AND DISASSEMBLING FAT OBJECTS
Partitioning a thin ob ject into two fat ob jects however is imp ossible We shall
prove that any split of an ob ject E that is not n fat for some large n results in
two parts of which one is not cn fat for some dimension dep endent constant
c b etween and In fact we even give a more general result stating that the
union of two intersecting ob jects cannot b e more than a constant factor less fat or
thinner than the least fat of its two constituents This statement do es not app ear
strange at all if we sup erimp ose two ob jects the result do es not seem to b e a lot
thinner than the original ob jects Before we prove the corresp onding lemma we
rst de ne a dimension dep endent constant The constant plays a role in the
d
up coming lemma
min ff j g
d
where
d
!
f
d d
d d
d
An analysis of the function f learns that it has a single minimum for
d
d
this minimum is reached at hence
d d
d
d d
f
d
p
Example values are and
! "
Now we are ready to formulate the lemma concerning the fatness of the union of
two intersecting fat ob jects
d d
Lemma Let E IR be a closed connected k fat object and let E IR be
!
a closed connected k fat object such that E E Then the union E E is
! ! !
max k k fat
!
d
Pro of The pro of obligation is that
max k k
!
volume E E S volume S S U
! E !E
!
d
or
d
S U volume E E S volume S
E !E !
!
max k k
!
Let us consider a randomly chosen S S U By de nition the center m
m r E !E
!
must lie in at least one of E and E Assume without loss of generality that m E
!
and recall that E S is the E enclosing hypersphere centered at m De ne r
m E ES
to b e the radius of E S so E S S We distinguish two di erent cases
m E m E m r
ES
r r or r r and analyze them separately starting with the rst and easiest
ES ES one
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
r r In combination with the connectedness of E the assumptions r r and
ES ES
m ! E yield S ! U Using the k fatness of E and we then get
E d
volume E " E # S
!
volume E # S
volume S
k
d
volume S
max k k
!
proving the inequality for all hyperspheres in U that do not fully contain
E E
!
E in their interior
r r In this case the hypersphere S has E completely in its interior The assump
ES
tion that S ! U then implies that the b oundary of S must intersect E
E E !
!
On the other hand the non emptiness of the intersection E # E and the
!
connectedness of E yield that E must also intersect the b oundary of E S
! ! m E
For the same pair of reasons nally E must also intersect the b oundary of the
!
hypersphere S which lies precisely halfway E S and S Now let
m E
m r #r !
ES
!
m ! E # S and de ne the hypersphere L S Note that
! m r #r ! m r "r !
ES ES
this hypersphere touches E S from the outside and S from the inside
m E
Because L and E S are disjoint and b oth contained in S we get that
m E
volume E " E # S
!
volume E " E # E S volume E " E # L
! m E !
volume E # E S volume E # L
m E !
E S is the largest hypersphere centered at m whose b oundary still inter
m E
!
sects E so E S ! U The hypersphere L has its center m in E and
m E E !
in addition its b oundary L is intersected by E b ecause E intersects the
! !
b oundaries E S and S which in turn b oth have non empty intersection
m E
with the interior of L Hence L ! U The memberships E S ! U and
E m E E
!
L ! U combined with the k fatness of E and the k fatness of E result in
E ! !
!
volume E # E S volume E # L
m E !
volume E S volume L
m E
k k
!
volume E S volume L
m E
max k k
!
r r & r
ES ES
d d
volume S volume S
max k k r r
!
The latter expression is simpli ed considerably by the de nition r r
ES
Note that by the assumption r r Substitution of and subsequent ES
ASSEMBLING AND DISASSEMBLING FAT OBJECTS
application of the de nition of yield
d
r r ! r
ES ES
d d
volume S volume S
max k k r r
!
!
d d
volume S volume S
max k k
!
d d
!
volume S
d
max k k
!
volume S
d
max k k
!
Combination of all inequalities gives
d
volume E # E $ S volume S
!
max k k
!
proving the inequality for all hyperspheres in U that completely contain
E E
!
E
The fact that S was randomly chosen from all hyperspheres in U centered in
E E
!
E and the symmetry of the construction with resp ect to E and E imply that the
!
inequality
d
volume S volume E # E $ S
!
max k k
!
holds for all hyperspheres S % U
E E
!
Informally Lemma states that if we place two fat ob jects in overlapping p osi
tions then the resulting union is not more than a constant factor less fat than the
least fat of the two united ob jects
Corollary is a generalized version of the result mentioned earlier and saying
that a thin ob ject cannot b e split into two relatively fat parts The result of the
corrolary is more general b ecause the ob ject is not decomp osed into two sub ob jects
but covered by two sub ob jects Note that a decomp osition is a restricted type of
covering in which the sub ob jects only overlap at their b oundaries Corollary is
essentially a reformulation of Lemma where k k k
!
d
Corollary Let E % IR be a closed connected object that is not k fat Any
covering of E by two closed connected parts results in at least one part that is not
k fat
d
Corollary supplies a crucial to ol for proving the main result of this section
Assume we are given an ob ject E that is not k fat Our aim is to partition it into or
! !
cover it by a preferably small number of k fat parts for some k k provided that
it is p ossible to do so Theorem rep eatedly applies Corollary to eventually
!
end up with k fat parts
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
d
Theorem Let E IR be a closed connected object that is not k fat and let
k k Any covering of E by k fat parts consists of log k k parts
Pro of Let P b e such that each P P is k fat and P E so P is a
P P
covering of E by k fat parts The union P is not k fat Now divide P into
P P
two non empty subsets P and P such that P and P are connected
! P P P P
!
sets By Corollary at least one of P and P is not k fat We
P P P P d
!
recursively apply the ab ove pro cedure to the subsets that contain more than one
d
log k k recursive set divisions to end up part As a result it takes at least
with subsets P P such that P is k fat Hence P must contain log k k
P P
parts
Theorem gives a lower b ound on the number of parts that is involved in any
covering of a thin ob ject by fat parts The existence of an upp er b ound on the
same number on the other hand seems unlikely Esp ecially for small values of k a
covering of the ob ject E which is not k fat by k fat parts may require many small
parts
Fatness de ned with resp ect to other shap es
This section presents a pro of of the supp osition from Section concerning the
close relation b etween fatness de nitions with resp ect to di erent compact shap es
that is if an ob ject is k fat with resp ect to a compact shap e A then it is k fat
with resp ect to some other shap e B for some k that is only a constant multiple of
k This supp osition sounds rather vague as it is yet unclear what exactly fatness
with resp ect to a shap e A means We de ne fatness with resp ect to A by rather
straightforward generalization of De nition
d
Let A IR b e a closed connected subset containing the origin O O A
"
Any scaled translate X of A can b e describ ed by X A m where IR is
d
a scaling parameter and m m m IR a translation vector So X
d
f a m a m j a a Ag
d d d
A A
U De nition U
E m E
d d
A
Let m IR and let E IR be an object The set U is de ned as
m E
A
f A m j A m E g U
m E
The set U is de ned as
E
A A
U U
m E E
m E
The following de nition is a generalization of the notion of fatness given in De nition
FATNESS DEFINED WITH RESPECT TO OTHER SHAPES
De nition k fatness with resp ect to shap e A
d
Let E IR be an object and let k be a positive constant The object E is k fat if
A
A
!S " U k volume E $ S volume S
E
Note that fatness with resp ect to some arbitrary shap e A is not invariant under rota
tion like the regular fatness with resp ect to hyperspheres Prop erty a c d
hold for the generalized notion of fatness
Theorem establishes a relation b etween the fatness of an ob ject with resp ect
to a shap e A and with resp ect to a shap e B The fatness of the ob ject E with resp ect
to B dep ends on its fatness with resp ect to A and the relative fatness of B with
resp ect to A
d d
Theorem Let A IR and B IR be shapes with O " A and O " B Let
d
E IR be a closed connected object If the object E is k fat and A itself is k fat
A B
then the object E is k k fat
B
B
the inequality k k volume E $ S Pro of We must prove that for each S " U
E
volume S holds We choose some arbitrary m " E In addition we choose some
arbitrary such that B m $ E & ' The set B m will b e denoted by
Y We de ne X A m such that is the largest p ositive real for which
d
X $ IR ( Y ' In words is the largest p ositive real for which X ts in Y
Note that p oint m not only lies in E but b ecause O " A and O " B also lies in
X and Y The intersection E $ X $ Y is therefore non empty See Figure for
a two dimensional example of the construction
A
The intersection of E and X Our rst step is to prove that X A m " U
E
is non empty Ob ject E will therefore either lie completely in the interior of X or
B
implies the b oundary X of X is intersected by E The assumption that Y " U
E
that the b oundary Y of Y is intersected by E As Y lies completely outside the
interior X ( X of X this means that the ob ject E must lie partly outside the
interior of X Combining the facts that the intersection of E and X is non empty
and that E lies partly outside the interior of X with the assumption that E is
connected implies that X $ E & ' Together with the fact that m " E we obtain
A
that X A m " U
E
The ob ject E is k fat which by de nition of fatness means that for each
A
A A
S " U k volume E $ S volume S Since X " U we yield
E E
k volume E $ X volume X
B
In a second step we prove that Y B m " U We know that m " X
X
so the only thing that remains to b e proven is that Y $ X & ' Assume for a
contradiction that Y $ X ' Since m " X $ Y this would imply that X lies
completely in the interior Y ( Y of Y But then we can grow X A m
by increasing while it remains tting in Y This ability to grow contradicts
CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
m E
X
Y
O O
B
A
Figure Construction of Y B m and X A m for some arbitrary
m E and some arbitrary establishing Y ! E " #
the assumption that is the largest p ositive real such that X m ts in Y
B
Hence we obtain that Y B m U
X
The ob ject A is k fat By the prop erty that the generalized fatness of an ob ject
B
is not a ected by translation and scaling of the ob ject the ob ject X A m
B
is also k fat By the de nition of fatness this means that for each S U
B
X
B
k volume X ! S volume S Since Y U and X Y we get
X
k volume X k volume X ! Y volume Y
Combining b oth inequalities with the straightforward inequality volume E !
Y volume E ! X induced by Y X results in the following lower b ound on
the part of Y covered by E
k k volume E ! Y k k volume E ! X k volume X volume Y
B
Recalling the fact that Y was chosen randomly from all members of U we may
E
conclude that in fact
B
k k volume E ! S volume S (S U
E
holds So the ob ject E is k k fat
B
A two dimensional example illustrates the use of the ab ove theorem Assume
we are given an ob ject E that is k fat i e E is k fat with resp ect to the unit
circle centered at O For some reason see for example Chapter our interest is
FATNESS DEFINED WITH RESPECT TO OTHER SHAPES
to determine what part of any axis parallel square with the intersection p oint of its
diagonals inside E and not fully containing E in its interior is covered by E In
other words we wish to know E s fatness with resp ect to the square C f x y
IR j x y g Now with the theorem available the only thing that remains
to b e done is to determine the fatness of the square w r t the circle With the
additional knowledge that the square is fat Theorem yields that E is k
fat and hence also that any axis parallel square with its diagonals intersecting in
C
E and not fully containing E is covered for at least k th by E
! CHAPTER FATNESS IN COMPUTATIONAL GEOMETRY
Chapter
Range searching and p oint
lo cation among fat ob jects
In this chapter we study two fundamental problems in computational geometry in
a context of fat ob jects p oint lo cation and range searching The p oint lo cation
problem aims at prepro cessing a set of disjoint geometric ob jects for e ciently re
p orting the sp eci c ob ject containing a query p oint The ob jective of the general
version of the range searching problem is to prepro cess a set of geometric ob jects
for quickly rep orting all ob jects intersecting some query range e g rectangloid
simplex hypersphere It is shown that arbitrary convex ob jects and or non convex
d
p olytop es in d dimensional space can b e prepro cessed in time O n log n log log n
d
into a data structure of size O n log n which supp orts p oint lo cation queries and
range searching queries with arbitrarily shap ed but b ounded size regions in time
d
O log n The data structure is based on Overmars structure for p oint lo cation
in fat sub divisions Let us brie y review some relevant results in b oth p oint
lo cation and range searching to place our result in a broader p ersp ective
Point lo cation in space has b een studied extensively and solved in a satisfac
tory way for many types of scenes as several solutions achieve logarithmic query
time and near linear storage after near linear prepro cessing time In
higher dimensional spaces on the contrary e cient solutions are available only for
!
restricted problem instances In space Chazelle obtains O log n query time
and an O n storage requirement for the case where the stored geometric ob jects
are the cells of a spatial sub division consisting of a total of n facets and satisfying
the restrictive constraint that the vertical dominance relation on its cells is acyclic
Preparata and Tamassia consider p oint lo cation in a set of disjoint convex p oly
"
hedra with total complexity n The p olyhedra sub divide IR into a number of
convex cells the p olyhedra and a single non convex cell the complement of the
!
p olyhedra Their data structure uses O n log n storage and is capable of answer
! !
ing p oint lo cation queries in time O log n after O n log n prepro cessing time
Go o drich and Tamassia improve the storage requirement for sets of disjoint
convex p olyhedra to O n log n without a ecting the query time The results for
CHAPTER RANGE SEARCHING AND POINT LOCATION
arbitrary dimension d are further restricted applying only to arrangements of hyper
planes or hypersurfaces of b ounded degree Clarkson presents a data structure
for p oint lo cation in an arrangement of n hyperplanes in d dimensional space The
d
structure supp orts queries in time O log n and requires roughly O n storage and
d
prepro cessing Chazelle and Friedman improve the storage to exactly O n at
!d"!
the cost of an increase of the prepro cessing time to O n Chazelle et al
achieve the same O log n query time for p oint lo cation in arrangements of hyper
!d #
surfaces of constant degree with a data structure of size roughly O n which is
!d !
computable in roughly O n time Apparently e cient solutions for sets of non
convex ob jects or non p olyhedra in space and for scenes other than arrangements
of hyperplanes or hypersurfaces of b ounded degree in higher dimensional spaces are
lacking
Nearly all pap ers on range searching discuss how to prepro cess a very elementary
class of geometric ob jects namely p oints for e ciently answering range search
queries with sp eci c range types The most extensively studied type of query range
is the orthogonal range and a long established result says see e g that a
d
set consisting of n p oints can b e prepro cessed in time O n log n into a data
d
structure of size O n log n which is capable of answering an orthogonal range
d
!
query in time O log n Some small improvements are p ossible Di erent
range types give rise to more complicated solutions In general the solutions to e g
simplicial range searching only provide low query time at the cost of a relatively
high storage requirement and prepro cessing time or vice versa At the one end one
d
nds solutions providing p olylogarithmic query time and roughly O n storage and
prepro cessing see whereas at the other end of the sp ectrum solutions
d
require only O n storage but guarantee only a larger query time of O n see
Van Kreveld gives similar b ounds for the problem of rep orting all
simplices that are entirely contained in a query simplex In b etween the previous
results are the trade o solutions which allow for exchanging storage for query time
An example of such a solution is given by Matou sek the presented structure
d"
d
supp orts queries in time O n m log m n at the cost of an O m storage
d
requirement where n m n Alternative trade o solutions provide similar
b ounds More sp eci c results with resp ect to dimension are rep orted by Chazelle
p
n log n query time and O n storage and Welzl who give a solution with O
!
! #
for triangular range searching in space and a solution with O n log n query
time and a storage requirement of O n log n storage for tetrahedral range searching
in space Semi algebraic query ranges are considered by Agarwal and Matou sek
resulting in a data structure of size O n with prepro cessing time O n log n
b "
which supp orts range queries with a region in d space in time O n where
is some arbitrarily small value and b is b ounded by d b d although its
c
All quoted rough b ounds are adequate up to a factor n log n for some arbitrarily small and
some constant c
!
In all quoted time b ounds we neglect the dep endence of the query time on the size of the
answer to the query All results actually have the form O f n h where h is the output size
precise value dep ends on
Overmars discusses a data structure for e cient and simple p oint lo cation
in fat sub divisions or sets of fat ob jects with total complexity n The structure
d d
supp orts p oint lo cation queries in time O log n and uses O n log n storage
In his pap er Overmars do es not touch the issue of e ciently computing the data
structure that is in time comparable to the storage requirement It is shown in
this chapter that for arbitrary convex ob jects and for non convex p olytop es the
d
structure can b e built incrementally in time O n log n log log n Besides sup
p orting e cient p oint lo cation queries the data structures storing the arbitrary
convex ob jects and or p olytop es can surprisingly also b e used for range search
ing with arbitrarily shap ed but b ounded size ranges In fact we show that each
b ounded size range query can b e implemented by a constant number of p oint lo ca
d
tion queries thus leading to a time b ound of O log n for range search queries
Chapter gives an imp ortant application of b ounded size range searching The mo
tion planning paradigm presented there requires the a priori knowledge of all pairs
of neighboring obstacles where the notion of neighboring is related to the size of the
smallest obstacle The results presented in this chapter facilitate the computation
d
of all pairs in time O n log n log log n
The main contribution of our results with resp ect to the p oint lo cation prob
lem lies in their dimensional generality where previous results in higher dimensions
are restricted to arrangements of hyperplanes or hypersurfaces of b ounded degree
and to severely restricted sub divisions and scenes of non intersecting convex p oly
hedra in space We nd that p oint lo cation queries in scenes of non intersecting
or mildly intersecting fat convex ob jects and or fat p olytop es can b e p erformed
in p olylogarithmic time at the cost of near linear storage and prepro cessing To
see the contribution for range searching we recall the results by Van Kreveld
quoted earlier A query with a simplex in d space for all contained simplices takes
d
logarithmic query time at the cost of roughly O n storage or linear storage at the
cost of p olynomial query time The solutions to range searching among p oints have
similar p erformance Although the range searching results apply only to fat ob jects
and small query ranges they succeed in combining p olylogarthmic query time with
near linear storage and query time for simplicial range searching among arbitrary
convex shap es and non convex p olytop es Moreover the data structure caters for
arbitrary query ranges
The rst section b elow discusses Overmars data structure for e cient p oint
lo cation among fat ob jects while the next section shows how the structure can
b e used for simple and e cient range searching among classes of fat ob jects In
the third section the range searching results are used to supp ort the incremental
construction of the multi purpose data structure starting from the largest ob ject
and rep eatedly adding the next largest ob ject Finally we summarize the results
and p oint out the various p otential generalizations
CHAPTER RANGE SEARCHING AND POINT LOCATION
Point lo cation among fat ob jects
This section discusses a data structure for p oint lo cation among disjoint fat ob jects
by Overmars The author presents the data structure as a structure for solving
the problem of p oint lo cation in sub divisions of d dimensional space into fat cells
The answer to the query is the sp eci c cell containing the query p oint The p oint
lo cation problem in fat sub divisions can b e seen as an instance of the following
more general formulation of the p oint lo cation problem
Given a set E of non intersecting constant complexity k fat ob jects in
d d
IR and a query p oint p IR rep ort the ob ject E E that contains p
or rep ort that no such ob ject exists
Figure shows two p oint lo cation queries in a set of ve non intersecting fat
ob jects The query with the p oint p should yield the answer E whereas the query
with q must result in the answer that no ob ject contains q If the ob jects in E
p
E
#
E
E
$
q
E
"
E
!
Figure Two p oint lo cation queries in the set fE E g the query with p
!
yields the answer E while q is rep orted to lie in no ob ject
d
entirely cover IR then we obtain a fat sub division like in Overmars pap er In the
more general setting the complement of the ob jects need not b e fat nor do es it
have constant complexity The ideas from are discussed b elow in the context of
this more general problem formulation
Overmars pap er only presents a data structure for e ciently answering p oint
lo cation queries the issue of building the structure remains untouched Later in this
chapter a solution for this problem is given for arbitrary convex ob jects and non
convex p olytop es The solution relies on the ability to do e cient range searching
queries among these ob jects Before discussing range searching and its application
to building the p oint lo cation structure this section simply summarizes the results
of Overmars presented in
Let us assume in this chapter that the constant complexity k fat ob jects in
E E E are ordered by radius of their minimal ob ject enclosing hyperspheres
n
POINT LOCATION AMONG FAT OBJECTS
Let furthermore b e the radius of E s minimal enclosing hypersphere Let us de
i i
note the axis parallel enclosing hypercub e of the minimal enclosing hypersphere of
an ob ject E by C By construction the hypercub es are ordered by increasing size
i i
Note that the side length of the hypercub e C is Notice that the hypercub es
i i
C may b e overlapping although the ob jects E are disjoint Furthermore let V b e
i i i
de ned as follows
V fE E j E C j ig
i j j i
Hence V is the set of ob jects that are larger than E i e with larger minimal
i i
enclosing hypersphere intersecting the b ox C Theorem immediately supplies
i
a useful prop erty of the sets V i n
i
Lemma For al l i i n jV j O
i
A second crucial lemma from is the following
Lemma Let p E and i min fhjp C g Then E C
j h j i
In words the lemma states that if the query p oint p lies in an ob ject E then E is
j j
an element of the set of ob jects V asso ciated to the smallest hypercub e C containing
i i
p This suggest the approach outlined b elow
Assuming that the hypercub es C and the sets V for all i n are available
i i
we pro ceed as follows to nd the answer to a p oint lo cation query with a p oint
d
p IR Determine the smallest hypercub e if any containing p If no hypercub e
contains p than p lies in no ob ject if on the contrary C is the smallest hypercub e
i
containing p then the set V must contain the answer to the query To this end we
i
check the ob jects in V for containment of p Note that the check for containment
i
of a p oint in an ob ject E V takes constant time due to the constant complexity
j i
of the ob jects The constant cardinality of V yields that the entire insp ection of
i
all ob jects in V takes O time If no ob ject in V contains p then no ob ject in
i i
E contains p otherwise the unique ob ject E V containing p obviously is the
j i
answer to the query As the insp ection of V takes constant time the p oint lo cation
i
query time is dominated by the time to nd the smallest hypercub e C containing
i
the query p oint p An appropriate data structure that solves this problem is given
b elow
The p oint lo cation problem is now essentially reduced to the following priority
p oint stabbing problem among intersecting hypercub es
d d
Given a set of hypercub es C in IR and a query p oint p IR rep ort the
smallest hypercub e C C containing p or rep ort that no hypercub e in
C contains p
To solve a priority p oint stabbing query among hypercub es Overmars prop oses a
d level data structure in which the upp er d levels are based on the segment tree
and the lowest level is a list or a balanced binary tree
CHAPTER RANGE SEARCHING AND POINT LOCATION
d The hypercub es C C are intervals on the real line The interval end
n
p oints partition the real line into n so called elementary intervals All
p oints within a single elementary interval are covered by exactly the same one
dimensional hypercub es We organize the intervals as an ordered list and lab el
each elementary interval with the index of the smallest of all one dimensional
hypercub es covering it The list structure requires O n storage The answer
to a p oint stabbing query is provided by the lab el of the elementary interval
containing the query p oint The interval can b e identi ed in time O log n
d The d dimensional hypercub es are stored in a segment tree T on their pro
jections onto the d th co ordinate axis The endp oints of the hypercub e pro
jections partition the d th co ordinate axis into a number of elementary inter
vals An interval I is asso ciated with each no de in T I is the union of
all consecutive elementary intervals asso ciated with the leaves of the sub
tree ro oted at With we store in an asso ciated structure the intersec
d
tions ! ! I # C for hypercub es C that entirely span the slab
i i
d d
! ! I but do not entirely span the slab ! ! I corre
!
sp onding to the parent of in T The pro jection of such an intersection
d
! ! I # C onto the subspace spanned by the rst d co ordi
i
nate axes is a d dimensional hypercub e We store these hypercub es in
a recursively de ned similar d level data structure on the rst d
co ordinates that is if d If d we use the one dimensional
construction outlined ab ove to store the intersection of the squares and the
planar slab The structure su ces b ecause the squares intersect the vertical
slab p erp endicularly and can therefore b e represented as intervals So the
b ottom level structure is a list or ordinary balanced binary tree instead of a
segment tree
Searching the multi level data structure with a query p oint p pro ceeds in the
following recursive manner start at the ro ot and rep eatedly continue towards the
child corresp onding to the slab containing p Testing only the last co ordinate is
su cient The search ends at the leaf corresp onding to the elementary interval
containing the last co ordinate We have now obtained O log n no des on the search
path each corresp onding to a slab containing p The search is continued recursively
in the substructures asso ciated to each of these no des The entire search from top to
d
b ottom in the multi level data structure takes therefore O log n time resulting in
d
O log n candidate answers The minimum among these candidates is the nal
answer to the query The query time can b e improved by applying fractional cas
cading to the two lower levels of the data structure This is p ossible b ecause the
b ottom level structures are ordered lists a sequence of intervals Fractional cascad
ing improves the query time in a level data structure consisting of a segment tree
!
with the one dimensional ordered lists as substructures from O log n to O log n
d
Hence a priority p oint stabbing query among hypercub es takes O log n time
RANGE SEARCHING BY POINT LOCATION
d
The structure uses O n log n storage The result is summarized in the following
theorem
d
Theorem A set E of non intersecting constant complexity k fat objects in IR
d
can be stored in a data structure of size O n log n such that for a query point
d d
p IR it takes O log n time to report the object E E that contains p or to
conclude that no object contains p
The remaining op en problem concerns the prepro cessing phase that is the
computation of the data structure given a set of non intersecting k fat constant
complexity ob jects E E ordered by increasing radii of their minimal enclosing
n
hyperspheres compute the multi level data structure storing the enclosing hyper
cub es C C of the minimal enclosing hyperspheres of these ob jects plus the
n
sets V V of larger ob jects intersecting the resp ective hypercub es C C
n n
d
Building the multi level data structure can b e accomplished in time O n log n
using standard techniques Another option that is exploited in Section is to
build the structure in an incremental way It is well known that a hypercub e can b e
d
inserted in a d level segment tree in time O log n Moreover if we use dynamic
fractional cascading instead of regular fractional cascading the insertion time
d
can b e further reduced to O log n log log n see Section
The computation of the sets V on the other hand seems to p ose more problems
i
Finding the ob jects E with j i intersecting the hypercub e C requires a range
j i
search query with C The query is not an ordinary range search query since we
i
are only interested in ob jects with a certain minimal size or index Performing
a range search query among all ob jects and subsequently ltering out the smaller
ob jects is not a go o d idea as the answer to the range query might by orders of
magnitude larger than V A b etter idea would b e to p erform a range search query
i
with V only among ob jects that are larger than E Surprisingly we show in the
i i
next section that it is p ossible in most interesting cases to use the p oint lo cation
structure itself for solving the range search query This suggests an approach where
we add the hypercub es from large to small meanwhile computing the sets V in
i
the following incremental way use b efore insertion of the hypercub e C the
n m
sets V V and the multi level data structure storing the hypercub es
n m! n m
C C to compute V by a range query with C among the ob jects
n m! n n m n m
E E Next insert C into the structure and continue with C
n m! n n m n m
Section contains the details of the approach
Range searching by p oint lo cation
In this section we use the p oint lo cation data structure to tackle the following general
version of the range searching problem
d
Given a set E of non intersecting constant complexity k fat ob jects in IR
with minimal enclosing hyperspheres with radii at least and a constant
CHAPTER RANGE SEARCHING AND POINT LOCATION
complexity query region R of arbitrary shap e with diameter at most h
for some p ositive constant h rep ort all ob jects E E that intersect R
Figure shows a b ounded size range query in a set of ve non intersecting fat
ob jects The query with the range R must yield the answer fE E E g Using
! "
E
#
E
E
$
R
E
"
E
!
Figure A range query in the set fE E g the query with R yields the
# "
answer fE E E g
! "
Theorem it is easy to verify that the answer to the query with a region R
satisfying the diameter b ound is a set of ob jects of constant cardinality Let us
de ne the set Q R of ob jects intersecting the region R
Q R fE E j E R g
It is shown how the p oint lo cation structure can b e used to solve the range
d #
searching problem in time O log n in the case that E is a set of arbitrary convex
ob jects and or non convex p olytop es The solution relies on lo cal prop erties of fat
ob jects The de nition of fatness requires a k fat ob ject E to have a large density
in the vicinity of any p oint p in the ob ject k th of a hypersphere centered at p
is covered by E It turns out that this prop erty makes it p ossible to hit any ob ject
or ob ject part with a certain minimum size regardless of its exact lo cation with at
least one p oint from a su ciently dense but not to o large pattern of sample p oints
whereas this would clearly b e imp ossible if the obstacle is non fat the chance to hit
a line segment by an extremely dense pattern of sample p oints is practically zero
To structure the problem and the shap e of its solution we restrict the sample p oints
to b e arranged as a regular orthogonal grid
De nition A regular orthogonal grid G r with resolution r is de ned by
G r f z r z r jz z Zg
# d # d
RANGE SEARCHING BY POINT LOCATION
This section fo cuses on the problem of nding a grid resolution such that a small
subset of the corresp onding regular orthogonal grid is guaranteed to hit any ob ject
E having non empty intersection with the query region R We shall rst determine a
suitable grid resolution for convex ob jects and subsequently use the results obtained
there to nd an appropriate resolution for general p olytop es
The main implication of these results is that the range query with the b ounded
d
size range R IR can b e solved by a sequence of p oint lo cation queries each taking
d
O log n time using the data structure for p oint lo cation among fat ob jects
Under the assumption that the diameter of the query region do es not exceed h
the sequence of p oint lo cation queries will have constant length
The aim is to nd a grid resolution r establishing that each ob ject E Q R is
hit by at least one p oint in some subset G r where preferably the size of
dep ends on k and h and the complexity of the individual ob jects only Before we
fo cus on the di erent types of ob jects we rst give some basic results that ease the
task to nd a grid resolution
To simplify the approach we will de ne for each ob ject a large hypersphere that
is contained in the ob ject E Q R For the hypersphere it is easy to determine
the grid resolution r such that the hypersphere is always hit by at least one of the
grid p oints
p
r d contains at least one point Prop erty Any hypersphere with radius at least
!
of the orthogonal grid G r
The hypersphere itself is determined in two steps First a result by Leichtwei
makes it p ossible to identify a large ellipsoid inside a convex part of the ob ject E
Due to the fatness of the ob ject E the ellipsoid in turn indeed contains a large
hypersphere
Ellipsoids play an imp ortant role throughout this section Let us therefore brie y
d
review some relevant prop erties of these shap es Any ellipsoid L IR can b e re
garded as a translated and rotated copy of an ellipsoid in so called standard p osition
An ellipsoid L in standard p osition has the following form
s
!
X
x
i
L
s
!
a
i
i f d g
i d i
where a a are constants The segment connecting the p oints a
d i
i d i
L and a L which has length a is referred to as an axis of L
s i s i s
L has d such axes If w a for all i d then the hypersphere with radius w
s i
centered at the origin is entirely contained in L see Figure The volume of an
s
ellipsoid can b e given as a function of the lengths of its axes the volume of L and
s
of its translated and rotated copies L is given by see e g
Y
volume L a
s d i
i f d g
CHAPTER RANGE SEARCHING AND POINT LOCATION
L
a
!
a
Figure The ellipse L with half axes a a contains a circle with radius a
! !
where is again the dimension dep endent constant multiplier from the volume
d
formulae for hyperspheres see Chapter
A lower b ound V on the volume of an ellipsoid L alone do es not su ce to prove
that L encloses a large hypersphere as it is p ossible to construct a very long and
thin ellipsoid An additional upp er b ound on the diameter of the ellipsoid and
hence on the a s i d however makes such a construction imp ossible This
i
follows easily from the volume formula for ellipsoids
d
be an el lipsoid with volume L V and let be an Lemma Let L IR
upper bound on its diameter Then L contains a hypersphere with radius at least
V
d
d
Pro of Assume without loss of generality that L is in standard p osition and hence
Q P
! !
a Then its volume is volume L a x of the form
i d
i f d g i f d g
i i
The upp er b ound of on the diameter of L implies that none of the axes of L is
longer than and thus for all i d
a
i
V
d
Now assume for a contradiction that a Subsequent application of
j
d
this inequality and the upp er b ound a i d yields
i
Y
a volume L a
i d j
i f d g i "j
Y
V
d
a
i d
d
i f d g i "j
V
d d
d
d
V
contradicting the assumption volume L V
V
d
The ellipsoid L entirely contains the hypersphere with radius cen
d
tered at the origin
RANGE SEARCHING BY POINT LOCATION
In the two subsections b elow we use the prop erty and lemma to nd a valid grid
resolution and identify a small subset of that grid that su ces to hit all ob jects
d
intersecting the query region R IR
Searching among convex ob jects
d d
Let E IR b e a convex k fat ob ject intersecting the query region R IR and
let m E R The hypersphere S b elongs to U as it is centered inside E and
m E
can imp ossibly have E entirely in its interior The membership S U and the
m E
k fatness of E yield
d
k volume E S volume S
m m d
The shap e E S is convex as it is the intersection of the convex ob jects E and
m
S The following result due to Leichtwei holds for any convex shap e
m
d d
I O
Lemma Let E IR be a convex object There exist el lipsoids L L IR
I O
such that L E L and
d I O
d volume L volume L
Corollary is a trivial consequence of Lemma
d
Corollary Any convex object E IR contains an el lipsoid L with
d
d volume L volume E
Application of Corollary to the shap e E S satisfying implies the
m
containment of an ellipsoid L E S such that
m
d
d
volume L
d
k d
The diameter of L is b ounded by b ecause L is contained in the hypersphere S
m
with diameter The application of Lemma to the ellipsoid L with diameter
at most and volume b ounded by yields that L contains a hypersphere S
d
with radius at least k d Prop erty subsequently implies that such a hyper
sphere is hit by at least one p oint from the regular orthogonal grid with resolution
d" d" d"
! ! !
or G k d Hence at least one p oint from G k d k d
hits S L E S
m
So far we have only b othered ab out nding a su ciently high resolution for a
grid to hit all ob jects E Q R Clearly it is unnecessary and even undesirable
to p erform p oint lo cation queries with a to o large subset of the grid b oth b ecause
it increases the query time and b ecause it would lead to many accidental hits of
ob jects E Q R Fortunately the size of the sample set and the number of accidental hits can b e adequately limited by a quick glance at the deduction of the
CHAPTER RANGE SEARCHING AND POINT LOCATION
grid resolution the resolution is chosen such that any ob ject E Q R is hit inside
a hypersphere S with m R This hypersphere lies entirely inside the region
m
R S where O is the origin of the Euclidean co ordinate frame and denotes
O
the Minkowski di erence op erator The Minkowski di erence of two sets A and B
is de ned by A B fa bja A b B g Hence at least one of the grid p oints
hitting E lies in R S As a result it su ces to restrict the set of sample p oints
O
to b e the set of grid p oints in the grown query region
d"
!
G k d R S
O
The result is summarized in the following lemma
R
R S
O
G
r
! G R S
r O
Figure A two dimensional example of the construction of the set of sample
p oints for a query with a region R with diameter h among a set E of ob jects
with minimal enclosing circle radius The resolution r of the orthogonal grid is
determined by the type of the ob jects in E
d
Lemma Let E be a set of convex k fat objects E IR with minimal enclosing
d
hyperspheres with radii at least and let R IR be a region with diameter h
for some constant h The set Q R of objects E E intersecting R can be found by
d"
!
point location queries with the points from G k d R S O
RANGE SEARCHING BY POINT LOCATION
The data structure presented in the previous section allows us to p erform p oint
d
lo cation queries among the ob jects of E with all p oints in in time O j j log n
The resulting set fE E j E g of query answers which clearly has at most j j
elements is a sup erset of the answer Q R to the range query with R For each of
the ob jects E f E E j E g additional constant time su ces to verify the
membership E Q R by a simple test for the non emptiness of E R provided
that R and all E E have constant complexity Hence the computation of Q R
d
takes O j j log n time
It remains to b ound the size of Clearly the Minkowski di erence R S
O
R R R R
ts entirely in the hypercub e x x h x x h
! ! d d
R
where x i d is the minimal x co ordinate o ccurring in R As a result the
i
i
number of elements in is b ounded by the number of grid p oints in the enclosing
hypercub e leading to
d#
!
j j jG k d R S j
O
d#
!
jG k d
R R R R
h j y h y x x
d d ! !
d d#
!
bhc k d
d d
O k d h
In a setting where all ob jects are k fat for some constant k and the diameter of the
query region R do es not exceed a constant multiple h of it follows that j j O
d
which implies an O log n time b ound for range searching with a b ounded size
region R
Theorem Let k and h be constants and let E be a set of convex k fat
d
constant complexity objects E IR with minimal enclosing hyperspheres with radii
d
at least A range searching query with a region R IR of diameter at most h
d
among E takes O log n time
Searching among p olytop es
Having solved the range searching problem among convex ob jects we now turn
d
our attention to non convex p olytop es We assume that all p olytop es E IR
in E are b ounded by c hyperplanar faces that is each face is part of a d
dimensional hyperplane Let E b e a k fat p olytop e intersecting R and let m E R
Inequality applies to the intersection of the p olytop e E and the hypersphere
S U on exactly the same grounds as in the convex case Unfortunately the
m E
intersection E S is not a p olytop e as its b oundary contains p ortions of the
m
hyperspherical b oundary of S This can b e remedied by replacing S by its
m m
axis parallel enclosing hypercub e C with center m and side length with m
CHAPTER RANGE SEARCHING AND POINT LOCATION
d
volume C volume S The ratio of the volumes the inequality
m m d
and the obvious inequality volume E C volume E S together yield
m m
d
k
d d
volume E C volume C
m m
d
The intersection E C is a collection of p olytop es The arrangement of the
m
c d supp orting hyperplanes of the c faces of E and the d faces of the hypercub e
d
C sub divide IR and more imp ortantly E C into convex regions the d faces
m m
or cells of the arrangement Edelsbrunner s b o ok on combinatorial geometry
supplies b ounds on the numbers of faces of various dimensions in arrangements of
hyperplanes Lemma repro duces the b ounds
d
Lemma The maximum number f n of k faces in an arrangement of n
k
d
hyperplanes in IR is given by
X
n d # i
d
n f
k
d # i k # i
i f " k g
We are interested in the maximum number of d faces in an arrangement of c d
d
d
hyperplanes in IR or f c d for short By Lemma we have that
d
X
d # i c d
d
f c d
d
d # i d # i
i f " d g
X
c d
j
j f " d g
X
c d
j
j f " c #$ dg
c#$ d %
The basis of the rst of the ab ove inequalities lies in the simple observation that
%
c#$ d %
d c d Note that the b ound of on the number of d faces is probably
$
not very tight as the c d hyperplanes include many parallel pairs of hyperplanes
c#$ d %
The c d hyperplanes sub divide the collection of p olytop es E C into g
m
convex regions The largest region E E C of these g convex regions clearly
m
satis es
volume E C volume E C volume E
m m
c#$ d %
g
d
d
c#$ d %
k
The convexity of the subshap e E E C allows for the subsequent appli
m
cation of Corollary Lemma and Lemma First of all Corollary tells
RANGE SEARCHING BY POINT LOCATION
us that the convex shap e E contains an ellipsoid L with
d
d
volume L
c ! d!" d
k d
As the ellipsoid L lies entirely inside the hypercub e C the diameter of L is
m
p
d Lemma now implies that L contains a hypersphere S with b ounded by
! c ! d!" !" ! % d!"
!
radius at least k d Prop erty nally shows that any such
hypersphere S is hit by at least one p oint from the regular orthogonal grid with
" "
d ! c ! d!! !" ! d ! c ! d!! !" !
! !
Hence at least one p oint from G k d resolution k d
hits S L E E C Notice that in the case of p olytop es unlike for convex
m
ob jects the required grid resolution dep ends on the complexity c of the ob jects of
E
By the considerations of the previous paragraphs one of the grid p oints that
hit any ob ject E Q R lies inside a hypercub e C with m E R This
m
hypercub e must therefore lie completely inside the Minkowski di erence R C
O
As a consequence the set may b e restricted to
"
! c ! d!! !" ! d
!
G k d R C
O
Lemma summarizes the results obtained so far in this subsection
d
Lemma Let E be a set of k fat polytopes E IR bounded by c hyperplanar
faces and with minimal enclosing hyperspheres with radii at least Furthermore
d
let R IR be a region with diameter h for some constant h The set Q R of
objects E E intersecting R can be found by point location queries with the points
"
d ! c ! d!! !" !
!
from G k d R S
O
Similar to the convex case the sequence of p oint lo cation queries with all p oints
d!"
in takes O j j log n time and results in the set fE E j E g The
extraction of Q R from this set takes O j j time under the additional assumption
that R and all E E have constant complexity so c must b e constant To b ound
the number of elements in we notice that R C also ts completely in the
O
R R R R R
i d h where x x h x x hypercub e x
i d d & &
is once again the minimal x co ordinate in R The number of grid p oints in the
i
hypercub e b ounds the number of elements in
"
d ! c ! d!! !" !
!
R C j j j jG k d
O
"
d ! c ! d!! !" !
!
jG k d
R R R R
x x h y y h j
& & d d
"
c ! d!! d d
!
k d bhc
c d d
O k d h
CHAPTER RANGE SEARCHING AND POINT LOCATION
If b esides c the parameters k and h are also constant then we get j j O
which induces p olylogarithmic query time for range searching with b ounded size
ranges among fat ob jects
Theorem Let k and h be constants and let E be a set of k fat
d
constant complexity polytopes E IR with minimal enclosing hyperspheres with
d
radii at least A range searching query with a region R IR of diameter at most
d
h among E takes O log n time
Building the data structure
The results of the previous section can b e used for the incremental construction of
the p oint lo cation and range searching data structure Let us assume we are given
the d level data structure for priority p oint stabbing queries among hypercub es from
Section storing the m largest hypercub es C C and the corresp onding
n m! n
constant cardinality sets V V We refer to this partial priority p oint
n m! n
stabbing structure as T Hence the ob jective is to eventually compute T from
m n
some initial structure T The outline of the incremental construction is as follows
"
compute T
"
m
while m n do
compute V by a range query with C
n m n m
using T and the sets V n m j n
m j
compute T by inserting C into T
m! n m m
We study b oth steps in the lo op in more detail starting with the second step as
the implications of its solution in uence the rst step as well
Computation of T
m!
The problem with the insertion of a hypercub e into the d level priority p oint stab
bing structure lies in the use of fractional cascading which was incorp orated to im
d d
prove the p oint lo cation and range search query time from O log n to O log n
Unfortunately insertions into the multi level data structure do not b ene t from
d
fractional cascading so an insertion into the structure T would require O log n
m
d
time instead of O log n Moreover a sequence of insertions into the multi level
data structure with the static fractional cascading part is likely to increase the time
d
for a query back to O log n as the fractional cascading part no longer suits the
up dated multi level data structure Building the data structure would even with
d
fractional cascading require O n log n time Fortunately Mehlhorn and N aher
describ e in a dynamic version of fractional cascading Incorp oration of dynamic
fractional cascading in the data structure during the construction phase improves
BUILDING THE DATA STRUCTURE
d
the prepro cessing time to O n log n log log n The alternative requires only mi
nor mo di cations We give the main result from in a formulation that is tailored
to our applications
Theorem Let T be a tree with t nodes and let an ordered list L of elements
from a given domain D be associated to each node Furthermore de ne l to be
P
the total length of al l lists L so l jL j
!
a Let T be a connected subtree of T with s nodes Location of a query value x
! ! ! !
in L for every T that is nding the position in L of the smal lest
value larger or equal than x takes O log l t s log log l t time worst case
b The deletion of a value x from a list L takes given x s position in L
amortized time O log log l t
c The insertion of a value x from a list L takes given the position in L
of the smal lest value larger than x amortized time O log log l t
To simplify the pro cess of incrementally building the ob jective structure T
n
we observe that all hypercub es that are to b e added throughout the construction
pro cess can b e computed in advance This observation facilitates a less complex
semi dynamic instead of dynamic prepro cessing The prior knowledge of all n
hypercub es means that the endp oints of the pro jections of the hypercub es on the
i th co ordinate axis are from a xed nite universe U of size O n We act however
i
as if we only know the pro jections of the hypercub es on the last d co ordinate axes
hence each corner x x x is assumed to b e a p oint from IR U U
! d ! d
The static nature of the hypercub es with resp ect to the last d axes is used
to compute the ma jor part of the d level data structure in advance by recursively
building a segment tree on the pro jections of the hypercub es onto the last d
co ordinate axes The resulting d level segment tree which can b e regarded as
the initial tree T in our incremental construction di ers from our ob jective d level
"
data structure T only in that the one dimensional ordered lists in the no des of
n
the substructures at level d are missing These lists are built incrementally
by inserting the hypercub es from large to small into the skeleton provided by the
d level segment tree Note that the substructures at level d represent
decomp ositions into vertical slabs of the plane spanned by the second and rst
co ordinate axes
Let us now consider the intermediate structure T obtained after inserting the
m
largest m hypercub es C C into the skeleton T Following the standard
n m# n "
insertion pro cedure for multi level segment trees the insertion of the hypercub e
C into T b oils down to the insertion of a square that is the pro jection of
n m m
d !
C onto the plane spanned by the rst two co ordinate axes into O log n
n m
substructures T at level d Note that the prior computation of the skeleton T
"
guarantees that no new no des have to b e created in any of the higher level structures
during the insertion of a hypercub e
CHAPTER RANGE SEARCHING AND POINT LOCATION
We move on to study the reduced problem of inserting the planar pro jection
L H L H
of C into a level d substructure T of T The upp er level i i i i
n m m
! !
of T is a segment tree on the pro jections onto the second co ordinate axis of all
at most n planar hypercub e pro jections stored in T The asso ciated ordered list
structure L of a no de of T stores the sequence from x to x
of disjoint slab intervals I within which all p oints share the same smallest
containing hypercub e The intervals are lab eled with the resp ective hypercub e index
The hypercub e C must b e stored in the ordered list substructures L at
n m
no des of T corresp onding to slabs IR I that are entirely spanned by the pro jection
L H L H
i i i i of C onto the plane spanned by the rst two co ordinate axes but
n m
! !
with parents parent corresp onding to slabs IR I that are not spanned by
parent
L H L H
i i i i Alternatively phrased the hypercub e C must b e stored in the
n m
! !
ordered lists L at no des corresp onding to intervals I that are entirely spanned
L H
by i i and have parents parent corresp onding to intervals I that are
parent
! !
L H
not spanned by i i Although the resulting no des do not form a connected
! !
subtree of T they are in fact never more than one no de o the search path from
R L
Hence we can or the endp oint i ro ot to leaf in T for either the endp oint i
! !
!
apply Theorem a to the connected subtree T of T consisting of all no des on
! ! !
and just o b oth search paths and therefore e ciently search all L for in T
L
of the pro jection of C simultaneously for the lo cation of the left endp oint i
n m
onto the rst co ordinate axis The search time dep ends see Theorem on the
!
number of no des s in the connected subtree T the cumulative length l of all
asso ciated ordered lists in T and the number of no des t in T First of all the tree
T is a priorly built tree on a subset of the pro jections of all hypercub es C C
n
!
so t O n The subtree T consists of two ro ot leaf paths in T plus all no des that
are only one no de o these search paths thus s O log n Moreover an ordered
list L b efore insertion of C stores only pro jections of the m hypercub es
n m
C C so jL j O m Because a part of each pro jection app ears
n m$ n
P
at no more than two no des at a single height level in T we have l jL j
!
O m log m O n log n Application of Theorem a to the subtree T of T
L
and the query value i yields that the lo cation of the query value is identi ed in
! ! !
all lists L for in T in time O log n log log n worst case Note that the set of
!
no des in T is a sup erset of the set of no des in whose asso ciated substructures C
n m
must b e inserted
L H
The problem that remains is to given the interval i i and p ointers to the
L ! L H
lo cation of i in all lists L with in T insert the interval i i only into
L H
the lists L of no des in which C must b e inserted i e i i spans I but
n m
! !
! L H
not I Let us consider a no de in T Verifying whether i i must b e
parent
L H
inserted into L is easily done in constant time by comparing i i with I and
! !
L H
I Assume that i i must indeed b e inserted into L lab eled with the
parent
L H
index n m After insertion of the interval a query with a p oint p i i I
for the smallest or lowest indexed covering hypercub e pro jection must obviously
L H
yield the answer n m Hence the interval i i must overwrite all parts of
BUILDING THE DATA STRUCTURE
L H
intervals that have non empty intersection with i i and are present in L
up on insertion of the latter interval Using the p ointer into L we can identify the
L H
subsequence of intervals that have non empty intersection with the interval i i
in time prop ortional to its length Let b e this sequence and note
g g
L H
that only may contain i and only may contain i The up date
g g
of L pro ceeds in four simple steps in which we scan all intervals intersected by
L H
i i
L L
if i then replace by i
else delete
for all h g " do delete
h h
H H
if i then replace by i
g g g g g
else delete
g g
L H
insert i i
As l O n log n and t O n the amortized time sp ent on each of the ab ove
deletions or insertions is by Theorem b c O log log n The four steps in
clude one insertion and g deletions in L The fact that some varying number of g
deletions take place during a single up date of a list L is not a problem due to the
observation that each deletion must follow an earlier insertion of the same interval
Hence at any time during the prepro cessing the number of insertions so far exceeds
the number of deletions So the amortized number of deletions p er list up date is
one as well which implies that the amortized time sp ent in up dating a single list
with a new hypercub e is O log log n Within each level d " substructure T the
required asso ciated list up dates are restricted to a subset of the O log n no des of the
subtree T so the time sp ent on all necessary list up dates in T is O log n log log n
Combined with the O log n log log n time b ound for the simultaneous search in all
L
this implies that the total lists L with in T for the lo cations of the value i
time sp ent on the up date of a single level d " substructure is O log n log log n
Since the total number of level d " substructures that have to b e up dated is
d!!
b ounded by O log n the insertion of C into T to obtain T takes time
n!m m m"
d!
O log n log log n
Lemma The insertion of the hypercube C into T to obtain T requires
n!m m m"
d!
O log n log log n amortized time
The computation of the structure T by inserting the hypercub e C into the
m" n!m
intermediate structure T is indep endent of the actual shap e of the ob jects under
m
consideration The e ciency of this part is therefore guaranteed irresp ective of the
ob ject shap e The e ciency of the computation of V however relies on the
n!m
fact that the ob jects under consideration are convex or p olytop es The dep endence follows from the use of the results from Section
CHAPTER RANGE SEARCHING AND POINT LOCATION
Computation of V
n m
The computation of V fE E j E C j n mg is based
n m j j n m
on a sequence of p oint lo cation queries In the static p oint lo cation structure
d
the query time was found to b e O log n due to the incorp oration of fractional
cascading Throughout the incremental construction of the p oint lo cation structure
however we use dynamic fractional cascading instead of fractional cascading to
d
achieve e cient insertions which leads to a query time of O log n log log n Let
us analyze a p oint stabbing query in the intermediate structure T
m
Like in the static case describ ed in Section a query with a p oint p in the
intermediate structure T pro ceeds recursively in the substructure asso ciated to
m
the O log n no des on the search path from ro ot to leaf This eventually leads
d !
to a search of O log n substructures at level d the level where dynamic
fractional cascading is incorp orated The upp er level of such a substructure T is
a segment tree on the pro jections onto the second co ordinate axis of the lo cally
stored hypercub es The query p oint p will therefore again b e contained in the
slabs corresp onding to the no des on the search path from the ro ot to the leaf of
the elementary interval containing p s pro jection Hence we must search all lists
L of no des on the search path Fortunately the no des on the path form a
!
connected subtree T of T with O log n no des so by Theorem a these no des
can b e searched simultaneously in worst case time O log n log log n The entire
d
p oint stabbing query time amounts to O log n log log n Note that a single search
d
with p yields O log n candidate answers one for each list L that is searched
The ultimate answer to the query is clearly the minimum among all hypercub e
indices found
The computation of the set V b ene ts from the fact that the hypercub es are
n m
inserted into the data structure from large to small in the sense that at the time of
the set s computation the intermediate data structure only stores hypercub es and
ob jects from the appropriate index range n m n Therefore we may
restrict ourselves to nding the hypercub es in the data structure that intersect the
query hypercub e C without having to b other ab out the sizes of these hypercub es
n m
Moreover note that future additions of hypercub es and their corresp onding ob jects
do not a ect the earlier computed sets V To apply the range searching results from
j
Section we must verify the validity of the constant ratio b etween the diameter
of the search region the hypercub e C and the lower b ound on the radii of
n m
the minimal enclosing hyperspheres of the stored ob jects The radii of the minimal
enclosing hyperspheres of the ob jects in fE E g are b ounded from b elow
n m" n
by and by the ordering on the radii also by The query region
n m" n m
C is the axis parallel enclosing hypercub e of the minimal enclosing hypersphere
n m
p
d The of E with radius As a result the diameter of C is
n m n m n m n m
application of Theorems and yields taking into account the mo di ed p oint
lo cation query time due to dynamic fractional cascading a worst case time b ound
d
of O log n log log n for the computation of V
n m
SUMMARY OF RESULTS AND EXTENSIONS
Lemma Let for al l n m j n V be a set of convex objects or polytopes
j
Then the computation of the set V from T and fV jn m j ng takes
n m m j
d
O log n log log n time
Lemmas and show that each of the O n steps in the incremental
construction of the d level data structure for p oint lo cation and range searching
d
among convex ob jects or p olytop es takes O log n log log n time resulting in a
d
time b ound of O n log n log log n for the computation of T from the skeleton
n
d
T Adding to this b ound the O n log n time b ound for building the d level
!
segment tree T we obtain the desired result
!
Theorem Let E be a set of non intersecting constant complexity k fat convex
objects or arbitrary polytopes Then the d level point stabbing structure can be built
d
in time O n log n log log n
After the construction of the data structure the query time can b e improved back to
d
n To this end it su ces to rebuild the structure using static fractional O log
cascading As all the sets V are now known this can easily b e achieved in time
i
d
O n log n
Summary of results and extensions
In this chapter we have presented a data structure for b oth p oint lo cation and range
searching with b ounded size ranges in certain scenes of fat ob jects Theorem
summarizes the results by combining Theorems and
Theorem Let k and h be constants and let E be a set of non
intersecting constant complexity k fat arbitrary convex objects and or non convex
d
polytopes E IR with minimal enclosing hypersphere radii at least Then the set
d d
E can be stored in time O n log n log log n in a data structure of size O n log n
which supports point location queries and range searching queries with ranges R
d
d
IR of diameter at most h among the objects of E in time O log n
The theorem do es not apply to scenes of arbitrarily shap ed non convex ob jects
It is though b elieved that a similar result holds in that case as well Preliminary re
sults in that direction with two dimensional ob jects b ounded by algebraic p olygonal
curves of b ounded degree are promising
Throughout the entire chapter the assumption that the ob jects in E are non
!
intersecting only plays a role in showing that the number of larger ob jects E inter
secting the enclosing hypercub e C of some ob ject E is b ounded by a constant No
other lemma or theorem relies on the disjointness of the ob jects As a consequence
all results in this chapter remain valid if we drop the requirement of disjointness
and instead imp ose the weaker restriction up on E that each enclosing hypercub e C
!
of E E is intersected by at most a constant number of ob jects E larger than E
CHAPTER RANGE SEARCHING AND POINT LOCATION
In the generalized setting of intersecting ob jects a query p oint may b e contained
in more than one ob ject The answer to a p oint lo cation or p oint stabbing query
should therefore b e the collection of ob jects containing the query p oint Note that
the new restriction on the data set E prevents more than a constant number of
simultaneous containments An interesting example of such a set is a collection E
w
of c wrappings of non intersecting k fat ob jects If the wrappings E E are
w
k fat for some constant k and convex or p olytop es then Theorem applies to
the set E of intersecting ob jects
w
The results of this chapter have an imp ortant application in the motion planning
part of this thesis The running time of the general paradigm for motion planning
amidst fat obstacles in Chapter dep ends on the time sp ent in computing the
pairs of obstacles within a distance b from each other where b is a constant
and is a lower b ound on the minimal enclosing hypersphere radii of the obstacles
in the workspace By Corollary there are only O n such pairs The result
d
in this chapter allow for the computation of all pairs in time O n log log log n
!
time instead of the trivial O n time As a related application it is p ossible to
compute the linear complexity arrangement by Theorem of tight wrappings
d
of non intersecting fat ob jects in time O n log log log n
Chapter
The complexity of the free space
In this chapter we return to the motion planning problem and start the investiga
tion on how fatness in uences the problem and its solution The problem of nding
a collision free motion for a rob ot in a workspace with obstacles is commonly trans
formed into the problem of nding a continuous curve in the free space In Chapter
we have argued that the complexity of nding such a curve highly dep ends on
the complexity of the free space This chapter studies the in uence of fatness on
the free space complexity More sp eci cally it shows that the complexity of FP
for a constant complexity rob ot moving amidst constant complexity fat obstacles is
linear in the number of obstacles provided that the size of the rob ot is prop ortional
to the size of the smallest obstacle and provided that the constraint hypersurfaces
de ned by the rob ot obstacle contacts are algebraic of b ounded degree
Section studies the structure of the free space in detail and establishes the
relation b etween the complexity of the free space and the number of multiple contacts
for the rob ot with the obstacles in the workspace Section gives an overview
of known results on free space complexities The overview gives an idea of the
conditions that typically lead to high complexities The observations are used in
Section to formulate a realistic framework of motion planning problems with a
linear complexity free space
The structure of the free space
The con guration space C is the space of parametric representations of all rob ot
placements The number of degrees of freedom f of the rob ot B determines the
dimension of the con guration space We classify the p oints in the con guration
space according to the rob ot placements that they represent resulting in three
di erent types of p oints Let Z C b e a placement of the rob ot B and let
B Z W b e the collection of p oints in the workspace covered by B when placed at
In the sequel we generally do not distinguish b etween the p oint Z C and the placement
that Z represents
CHAPTER THE COMPLEXITY OF THE FREE SPACE
Z Furthermore we denote the interior of a closed set X by int X Assume that
Z C then
Z is a free placement when B Z E
E E
Z is a contact placement when B Z E and B Z int E
E E E E
Z is a forbidden placement when B Z int E
E E
Informally a free placement is a placement in which B do es not intersect any obsta
cle a contact placement is a placement in which B is in contact with the b oundary
of some obstacle but do es not intersect the interior of any obstacle and a forbid
den placement is a placement in which B intersects the interior of some obstacle
Clearly any p oint Z C satis es exactly one of the three expressions and corre
sp onds therefore to either a free placement or a contact placement or a forbidden
placement
The subset of con guration space of all free placements can according to the
ab ove classi cation b e obtained by subtracting the union of all sets C fZ
E
C jB Z E g with E E from C FP C n C A set C is sometimes
E E E E
see e g referred to as a con guration space obstacle of E as it consists of
all placements of B in which it intersects E The b oundary C consists of rob ot
E
placements Z such that B Z E and B Z int E or in other words of
all placements in which B touches E The set of placements C in which B touches
E
E separates the placements of C in which B intersects the interior of E from the
E
placements of C n C in which B do es not intersect E On a more global level
E
we nd that the union b oundary C separates the forbidden placements
E E E
from the free placements The union b oundary equals exactly the set of all contact
placements Notice that the semi free space SFP de ned in Chapter consists of
all free placements and all contact placements
The b oundary C of a con guration space obstacle C in the f dimensional
E E
con guration space can b e regarded as a collection of f dimensional hypersur
faces consisting of contact placements of a single rob ot feature and a single obstacle
feature of appropriate dimension We use the term feature to describ e a basic part
of the b oundary of a geometric ob ject whether an obstacle or the rob ot An f
dimensional hypersurface of contact placements is called a constraint hypersurface
The lower dimensional features on the b oundary of a con guration space obstacle
are common b oundaries or intersections of two or more constraint hypersurfaces
For example in the two dimensional con guration space C IR of a translating
p olygonal rob ot amidst p olygonal obstacles each constraint curve is induced either
by the contact of a rob ot vertex with an obstacle edge or by the contact of a rob ot
edge with an obstacle vertex Figure shows b oth types of contact If the rob ot is
allowed to rotate as well then b oth combinations de ne constraint surfaces in the yet
three dimensional con guration space C IR In the three dimensional
!
con guration space C IR of a translating p olyhedral rob ot amidst p olyhedral
THE STRUCTURE OF THE FREE SPACE
B
B
E
E
Figure The two types of contacts inducing a constraint curve in the con guration
space of a p olygonal rob ot translating amidst p olygonal obstacles a rob ot vertex
sliding along an obstacle edge and a rob ot edge sliding along an obstacle vertex
obstacles each constraint surface is induced by the contact of either a rob ot vertex
and an obstacle face or a rob ot edge and an obstacle edge or a rob ot face and an
obstacle vertex
The constraint hypersurfaces in con guration space allow for an interpreta
tion of the free space with a more computational geometry like avor The f
dimensional constraint hypersurfaces partition the f dimensional con guration
space into f dimensional cells A cell is a maximal connected f dimensional subset
of the con guration space containing no part of a constraint hypersurface The cells
consist either exclusively of free placements of exclusively of forbidden placements
The cells are referred to as free cells and forbidden cells resp ectively The free cells
in the arrangement of constraint hypersurfaces collectively constitute the free space
FP We are therefore interested in studying a collection of cells namely the free
cells in the partitioning of f dimensional space by a collection of f dimensional
hypersurfaces Note that by the de nition of a cell no two p oints in two di er
ent free cells are linked by a free path that is a path that is entirely contained in
the free space The de nition of the free space via the arrangement of constraint
hypersurfaces links the study of the motion planning problem to a basic study in
computational geometry namely the study of arrangements of hypersurfaces
Before we de ne the complexity of a cell in an arrangement we rst formulate
an assumption regarding the constraint hypersurfaces in con guration space The
assumption stands throughout all of the remaining chapters
The hypersurface in con guration space corresp onding to the set of place
ments in which a certain rob ot feature is in contact with a certain ob
stacle feature of appropriate dimension is algebraic of b ounded degree
The assumption on the shap e and complexity of the constraint hypersurfaces mainly
means that the b oundaries of the rob ot and the obstacles are not to o irregularly
CHAPTER THE COMPLEXITY OF THE FREE SPACE
shap ed A direct consequence of the assumption is that the intersection of any mul
tiple of hypersurfaces consists of only a constant number of connected comp onents
f
Moreover it implies a rst simple upp er b ound of O n on the complexity of the
entire arrangement of constraint hypersurfaces
The complexity of a cell in an arrangement of algebraic hypersurfaces of b ounded
degree is de ned to b e the number of faces of various dimensions on the cell s b ound
ary A j dimensional face or j face is a maximal connected j dimensional part of the
arrangement containing no lower dimensional faces in its interior A j dimensional
face of the sp eci c arrangement of constraint hypersurfaces is a maximal connected
comp onent of the intersection or common b oundary of f j constraint hypersur
faces For example the complexity of a two dimensional cell in an arrangement of
line segments in the plane is the number of edges faces and vertices faces on
the cell s b oundary where a vertex is either an endp oint of a line segment or the
intersection p oint of segments and an edge is a maximal p ortion of a line segment
meeting no vertex of the arrangement
The complexity of the free space is the sum of the complexities of the free cells
and hence b ounded by the complexity of the entire arrangement of constraint
hypersurfaces that is the total number of faces of any dimension in the arrangement
As each constraint hypersurface is induced by a contact of a rob ot feature and an
obstacle feature the intersection of j such surfaces corresp onds to the simultaneous
o ccurrence of j contacts for the rob ot Because each intersection of j hypersurfaces
consists of a constant number of connected comp onents by the assumption on the
shap e and complexity of the hypersurfaces the number of j fold contacts is of the
same order of magnitude as the number of j dimensional faces in the arrangement
Thus the complexity of the free space is determined by the total number of di erent
single and multiple contacts for the rob ot since they determine the complexity of
the arrangement of constraint hypersurfaces which b ounds the complexity of FP
To get a feeling of what a multiple contact is consider the case of a ladder
line segment translating among p olygonal obstacles in the plane This is a mo
tion planning problem with two degrees of freedom and the constraint curves that
it induces in the con guration space C IR are straight line segments Each of
these constraint segments is induced either by the contact of a ladder endp oint with
an obstacle edge or by the contact of the interior of the ladder with an obstacle
vertex Consider now the case where each ladder endp oint touches a distinct obsta
cle edge and assume further that these two edges have di erent directions The
contact of each ladder endp oint with an obstacle edge is expressed as a segment in
the con guration space and this double contact will manifest itself as the meeting
p oint of these two segments namely as a vertex in the con guration space The
fact that the double contact o ccurs at an isolated p oint in con guration space can
b e easily understo o d by observing that it is imp ossible to maintain the double con
tact while slightly moving the ladder If the ladder is also allowed to rotate then
the single contacts mentioned ab ove de ne constraint surfaces in the con guration
space IR The double contact of the rob ot in which its two endp oints touch
THE STRUCTURE OF THE FREE SPACE
two di erent non parallel edges now de nes a curve in the con guration space the
intersection of the two constraint surfaces corresp onding to b oth contacts of the
endp oints The fact that the double contact de nes a curve can b e understo o d by
the observation that it is p ossible to slide b oth rob ot endp oints along the resp ective
obstacle edges that they touch thus maintaining the double contact The continu
ously changing rob ot placements lie on a curve in con guration space An additional
third contact for the ladder rob ot for example when its interior touches an obstacle
vertex xes the p osition of the rob ot in the sense that is unable to move without
losing at least one of the three contacts Triple contacts therefore o ccur at isolated
p oints in con guration space they corresp ond to intersections of three constraint
surfaces
In the preceding paragraphs we have implicitly assumed that the rob ot can only
collide with the obstacles and not with itself In other words we have assumed that
no part of the rob ot can collide with another part of the rob ot Although the absence
of such so called self collisions or self intersections is a common assumption in
motion planning we choose to give some thought to the p ossible consequences when
self collisions are not neglected Self collisions clearly only o ccur when the rob ot
under consideration is not a single rigid b o dy but instead consists of a number of such
b o dies linked together by revolute or prismatic joints A sp eci c prop erty of self
collisions is that they dep end solely on the relative p ositions of the rob ot parts the
lo cation of the rob ot in the workspace is irrelevant for determining if a certain rob ot
placement causes self collision Figure illustrates the observation Consider the
J
Figure An example of a self colliding DOF rob ot
rob ot consisting of a triangle and a line segment attached to each other by a revolute
joint J the triangle edges incident to J de ne an angle The joint J serves also
as the rob ot s reference p oint A placement of this DOF rob ot is sp eci ed by the
p osition x y of the reference p oint in the workspace W IR and by the angles
and b etween the p ositive x axis and the line segment and the triangle resp ectively
Clearly the two linked parts of the rob ot intersect in any placement x y
satisfying mo d Hence the entire subspace IR f
j mo d g consists of forbidden placements due to
CHAPTER THE COMPLEXITY OF THE FREE SPACE
self intersections of the rob ot Rob ot rob ot collisions can b e dealt with in exactly
the same way as the rob ot obstacle collisions so that we end up with a number of
self collision constraint hypersurfaces that separate placements in which parts of the
rob ot intersect each other from placements in which the rob ot do es not self intersect
A self collision constraint hypersurface is induced by the contact of a rob ot feature
with another rob ot feature of appropriate dimension The self collision constraint
hypersurfaces in the linked DOF rob ot example are the two three dimensional
surfaces and The arrangement of all regular and self collision
constraint hypersurfaces sub divides the f dimensional con guration space in free
and forbidden cells w r t the obstacles and the rob ot itself Below we search for
conditions for the self collisions that prevent an increase of the complexity of the
arrangement and hence of the complexity of the free space
The shap e of the self collision constraint hypersurfaces di ers somewhat from
the shap e of the regular constraint hypersurfaces in the sense that they are larger
The b ounded range of reference p oint p ositions in the workspace in which a sp eci c
rob ot feature touches a sp eci c obstacle feature is re ected in a certain compactness
of the corresp onding constraint hypersurface in con guration space On the con
trary a collision of two rob ot features is indep endent of the p osition of the rob ot s
reference p oint so that the corresp onding constraint hypersurface can b e unbound
edly large see for example the surfaces of the ab ove example The size of the
self collision constraint hypersurfaces causes such surfaces to p ossibly intersect all
other surfaces If however the number of self collision constraint hypersurfaces is
constant and each hypersurface is algebraic of b ounded degree then certainly these
additional surfaces will not increase the asymptotic complexity of the arrangement
of constraint hypersurfaces Because of our general assumption that the rob ot is of
constant complexity this is always the case Hence in b ounding the free space com
plexity we may neglect self collisions We brie y revisit self collisions in Chapter
to examine their algorithmic consequences
Unfortunately the worst case number of multiple contacts and hence the com
plexity of the free space can b e high If n is the number of obstacle features and
f is the number of degrees of freedom of the rob ot i e the dimension of the con
guration space and the number of rob ot features is b ounded by some constant
f
then this complexity can b e n So theoretically motion planning techniques
whose p erformances dep end on the size of the free space are exp ensive Fortunately
in many practical situations the complexity of the free space FP tends to b e much
smaller and as a result such metho ds might b ecome feasible A study of prop er
ties that limit the number of multiple contacts for the rob ot and consequently the
complexity of FP is therefore of obvious imp ortance
In many practical cases the relative p ositions and the shap es of the obstacles are
such that the number of multiple contacts for the rob ot B is very low Obstacles
that lie far apart clearly result in less multiple contacts for B than obstacles that
are cluttered Similarly obstacles that have long and skinny parts will induce more
RESULTS ON FREE SPACE COMPLEXITIES
multiple contacts than fat obstacles that do not have such parts Before we for
malize these intuitions in the Section we rst review some known b ounds on the
complexities of the free space for motion planning problems
Results on free space complexities
Research in motion planning throughout the last few years has concentrated on
studying the structure and complexity of the free space rather than on developing
new motion planning algorithms This trend can b e explained from the fact that a
thorough insight in the structure and complexity of the free space is evidently very
imp ortant for the design of algorithms that e ciently prepro cess FP into a small
structure capable of providing fast answers to motion planning queries
Besides studying the entire free space several pap ers also fo cus on the complexity
and computation of a single connected comp onent or single cell of the free space
The motivation for this direction of research lies in the simple observation that a
rob ot can only reach placements that lie in the free space comp onent containing the
initial rob ot placement The complexity of a single free cell is sometimes an order
of magnitude smaller than the complexity of the entire free space
In the preceding section we have seen that the complexity of an arrangement
of O n constraint hypersurfaces in f dimensional con guration space is b ounded
f
by O n by standard arguments on arrangements of algebraic hypersurfaces of
b ounded degree For certain motion planning problems that is for certain rob ots
and obstacle types the shap es of the constraint hypersurfaces are such that the
arrangement of hypersurfaces and hence the complexity of the free space has a
f
worst case complexity smaller than O n Given the upp er b ounds obtained by
combinatorial arguments it is interesting to see if examples of motion planning
environments that is a rob ot in a workspace with obstacles can b e constructed
that do indeed achieve these worst case free space complexities When trying to do
so it turns out that it is often di cult to construct settings of rob ots in workspaces
with obstacles that establish or even approximate the upp er b ound on the free
space complexity or single cell complexity thus leaving a gap b etween theoretical
worst case b ounds obtained by combinatorial arguments and complexities obtained
by constructed di cult complex motion planning environments The existence of
such a gap clearly raises uncertainty on the tightness of the upp er b ound
The lower b ound constructions of di cult motion planning environments found
in literature are often very arti cial and as such rarely encountered in real life situ
ations they often involve rob ots and obstacles that are extremely thin and or have
exorbitant relative sizes If the extreme prop erties of the rob ot and the obstacles
that are necessary to construct these di cult settings do not o ccur then most of
the arti cial lower b ound constructions b ecome imp ossible This illustrates that the
complexity of the free space for practical motion planning problems is likely to stay far b elow the worst case complexities obtained by combinatorial arguments and ap
CHAPTER THE COMPLEXITY OF THE FREE SPACE
proximated by arti cial settings Below we review some of the known lower b ound
constructions and upp er b ounds on free space and single cell complexities for motion
planning problems The currently available results are restricted to problems with
at most three dimensional con guration spaces We consider motion planning prob
lems involving a constant complexity rob ot amidst n constant complexity obstacles
The trivial upp er b ound on the complexity of the free space of a translating
p olygonal rob ot f amidst p olygonal obstacles is O n If the rob ot is convex
then the worst case complexity of the free space remains an order of magnitude
b elow the trivial b ound see e g The linear b ound is clearly optimal Things
change if the rob ot is allowed to b e non convex Figure shows an L shap ed
rob ot amidst n vertical line segments close to each other and n horizontal
line segments arranged similarly on a horizontal line If the rob ot s horizontal bar
n
B
n
Figure A planar translational motion planning problem with free space com
plexity n
is placed b etween two consecutive horizontal line segments and its vertical bar is
placed b etween two vertical segments then b oth bars of the L are stuck b etween
these two pairs of segments and the reachable p ositions of the vertex incident to
the two bars are restricted to a small dotted square of p oints These p osition
constitute a separate connected comp onent of the free space As there are n
combinations of a vertical and horizontal segment the free space consists of n
free cells Hence the complexity of FP is n Let us try to nd out what
sp eci c prop erties of the construction in Figure lead to the quadratic free space
complexity First of all it turns out that a similar constructions can b e obtained
for any combination of sizes of the rob ot and the segments simply by appropriately
choosing the distance b etween any pair of consecutive parallel segments Thus a
restriction on the relative sizes alone do es not make the construction imp ossible
On the other hand a minimal fatness restriction on the obstacles alone is also
insu cient In the case that the line segments are replaced by squares of equal size
RESULTS ON FREE SPACE COMPLEXITIES
the ab ove construction can b e obtained by choosing the rob ot su ciently large More
obstacles though necessitate a larger rob ot so the size of the rob ot is prop ortional
to the number of obstacles This observation indicates that the combination of a
minimal fatness requirement and a b ound on the relative sizes of the rob ot and the
obstacles make the construction imp ossible The results in the next section con rm
the supp osition For the sake of completeness we mention that the complexity of
a single free cell for a p olygonal rob ot amidst p olygonal obstacles is O n n
where n is the extremely slowly growing inverse of the Ackermann function
The complexity of the free space of a translating and rotating p olygonal rob ot
f amidst p olygonal obstacles is trivially b ounded by O n Leven and Sharir
rep ort an O n n b ound on the complexity if the rob ot is a convex p olygon
!
where n is a near linear function dep ending on the length of certain so called
!
Davenport Schinzel sequences For a discussion of Davenport Schinzel sequences
and more detailed b ounds on n for various values of s the reader is referred
s
"
to Ke and O Rourke show that n moves that is constant complexity
curves may b e necessary to connect two placements in a single free cell This
result gives an indication of the p otential complexity of nding a path in a single
cell the cell complexity alone do es not give full insight in this matter nding a
path b etween two p oints in a convex cell for example is simple regardless of its
"
complexity Figure gives an n lower b ound construction for the single cell
complexity and hence for the complexity of the entire free space that approximates
the near quadratic upp er b ound of O n n thus leaving a relatively small gap
!
b etween the theoretical upp er b ound and achievable lower b ound construction The
n
B
n
Figure A planar motion planning problem for a convex rob ot with free space
"
and single cell complexity n
rob ot B which is a simple line segment moving among small p oint or square
obstacles can b e in simultaneous contact with any combination of obstacles from
"
the top and b ottom row yielding n obstacle pairs Any simultaneous contact
with features of such a pair de nes a one dimensional face on the b oundary of the
"
free space resulting in n one dimensional faces All double contact placements
CHAPTER THE COMPLEXITY OF THE FREE SPACE
are connected by semi free paths and b elong therefore to the b oundary of a single
connected comp onent of the free space Thus the complexity of this free cell and
obviously also of the entire free space is n Like in the purely translational case
it is p ossible to build the construction if the obstacles have a certain minimal fatness
simply by making the rob ot su ciently large A restriction on the relative sizes of
the rob ot and the obstacles however seems to make the construction imp ossible
Once more the combination of b oth assumptions implies a linear upp er b ound on
the complexity of FP as will b e shown in Section
!
Contrary to the case of a convex rob ot the trivial upp er b ound of O n on
DOF motion planning can b e achieved for some construction involving a non convex
rob ot Figure shows an L shap ed rob ot moving among two horizontal rows of n
obstacles and one vertical row of n obstacles The example is taken from a pap er
by Halp erin Overmars and Sharir on motion planning for an L shap ed rob ot
For appropriately chosen distances b etween the three rows and b etween consecutive
n
n
B
n
Figure A planar motion planning problem for a non convex rob ot with free
!
space complexity n
obstacles within a single row it can easily b e veri ed that it is p ossible to place
one bar b etween any combination of pairs of consecutive obstacles in the upp er and
lower horizontal rows and the other bar b etween any pair of consecutive obstacles in
the vertical row Once the bars are placed b etween consecutive pairs of obstacles in
each of the three rows the rob ot is stuck b etween these pairs it is not connected
by a free path to a placement of the bars b etween di erent obstacle pairs As the
! !
number of combinations is ab out n the free space consists of n free cells
!
and has complexity n Like in the previous examples the construction can b e
built for fat obstacles as well A restriction on the relative sizes of the rob ot and
RESULTS ON FREE SPACE COMPLEXITIES
the obstacles probably makes the construction imp ossible the worst setting that is
then achievable seems to b e the setting of Figure yielding quadratic size free
!
O log n
space Halp erin and Sharir present an upp er b ound of O n on the
complexity of a single connected comp onent of the free space This b ound is almost
tight as a quadratic lower b ound construction is given by Figure if we apply a
minor mo di cation to the rob ot to turn it into a non convex shap e see also
Finally we brie y consider a translating p olyhedral rob ot f among p oly
hedral obstacles in a three dimensional workspace with a trivial upp er b ound of
O n on the complexity of the free space A rather straightforward generalization
of the construction of Figure shows that the cubic complexity can indeed
by achieved if the rob ot is non convex Figure shows the three dimensional con
n
n
B
n
Figure A spatial translational motion planning problem for a non convex rob ot
with free space complexity n
struction the orientations of the three sets of parallel planes restrict the p osition
of the meeting p oint of the three bars of B to a small cub e similar to the planar
example The construction b ecomes imp ossible up on addition of a minimal fatness
requirement for the obstacles for reasons similar to those of the corresp onding pla
nar case Other results on the complexity of the free space are restricted to convex
rob ots Results by Wiernik and Sharir imply the existence of a lower b ound
construction of size n n for a translating convex p olyhedral rob ot among
p olyhedra based on Davenport Schinzel sequences Halp erin and Yap prove an
upp er b ound of O n n for the free space complexity of a translating b ox con
rming the more general conjecture of Sharir that the same upp er b ound holds
for any convex p olyhedral rob ot Recently Aronov and Sharir have shown that
the worst case complexity of FP for a convex p olyhedral rob ot amidst p olyhedral
obstacles is O n log n Halp erin and Sharir nally prove an upp er b ound on
CHAPTER THE COMPLEXITY OF THE FREE SPACE
the complexity of a single cell in an arrangement of n low degree algebraic surface
patches in space of O n for any where the constant of prop ortionality
dep ends on The result b ounds the complexity of a single connected comp onent of
the free space for motion planning problems with three degrees of freedom involving
a rob ot and obstacles of constant complexity
Ke and O Rourke rep ort a lower b ound for a more complicated spatial motion
planning problem namely that of a translating and rotating ladder f among
p olyhedral obstacles They show that n distinct moves or constant degree
curves in con guration space may b e necessary to connect two placements of the
ladder It is not hard to understand that n moves can only b e necessary within
a free cell with at least the same complexity n and hence also in a free space
with complexity n
Fat obstacles and the free space complexity
In this section we deduce a linear b ound on the complexity of the free space for a
rob ot moving amidst fat obstacles and formulate the necessary additional assump
tions that lead to the result The considerations in the rst section of this chapter
show that the number of multiple contacts is of the same order of magnitude as the
complexity of the free space provided that the constraint hypersurfaces are algebraic
of b ounded degree Under this assumption we may therefore settle for a b ound on
the number of multiple contacts to b ound the complexity of the free space A very
useful observation now is that two obstacle features that are far apart more than
the maximum diameter of the rob ot cannot b e involved in any multiple contact for
B simply b ecause B is unable to touch b oth simultaneously Hence obstacles that
do induce such a contact must lie in each other s proximity
As the notion of proximity dep ends on the size of the rob ot we must rst de
termine a convenient way of expressing this size The fact that the rob ot may b e
articulated implies that its diameter is variable so a more general notion is needed
Let O B b e the rob ot s reference p oint The reach of a rob ot B is de ned to
B
b e the maximum distance from the reference p oint O B to any p oint in B in any
placement Z of B More formally
De nition reach of a rob ot B
B
Let Z be some arbitrary position of the reference point O of the robot B Then the
W
reach of the robot B is de ned as
B
sup max d p Z
B W
p B Z Z
W D
Z D
D
In words the reach of a rob ot B is the maximum distance in the workspace that
B
any p oint in the rob ot B can ever have to the reference p oint which is also equal
to how far the rob ot can reach measured from its reference p oint Naturally the
reach is indep endent of the actual p osition of the reference p oint The de nition
FAT OBSTACLES AND THE FREE SPACE COMPLEXITY
involves a sup instead of a max b ecause the rest space D might b e op en like e g
D in the case of the planar rotating and translating rigid rob ot Notice
that the de nition of the reach causes any rob ot with reach and its reference
B
p oint O placed at p W to b e completely contained in the hypersphere S with
p
B
radius centered at p regardless of the actual placement Z of the rob ot Note
B
furthermore that the maximum diameter of a rob ot with reach is In the
B B
remainder of the thesis the reach of the rob ot will b e used as the main means of
expressing the rob ot size
A convenient strategy for b ounding the number of multiple contacts is by charg
ing each multiple contact to the smallest obstacle involved in the contact and sub
sequently b ounding the number of chargings to any obstacle E The observation in
the previous paragraph learns that all features involved in a contact of B with E
and larger obstacles must lie in the proximity of E Corollary supp orts this
strategy by supplying a valuable b ound on the number of obstacles and hence on
the number of features of obstacles larger than E that lie in E s proximity The
corollary which we recall b elow as Prop erty in a form that is tailored to our
current needs also gives clear indications on what additional assumptions on the
workspace W and obstacles E are required for obtaining a linear number of multiple
contacts
Prop erty Let k and b be constants and let E be a set of non
d
intersecting k fat objects in IR Let E E be an object with minimal enclosing
hypersphere radius Then the number of objects E E with larger minimal en
closing hypersphere radii within a distance b from E is bounded by the constant
d
k b
The imp ortance of the prop erty b ecomes clear if we realize that a rob ot with reach
b can only simultaneously touch obstacles that are less then b
B B
apart The features involved in a multiple contact of B with minimal enclosing
hypersphere radius b with E and larger must clearly b e among the features
B
d
of the at most k b O obstacles in the proximity of E
Before we fo cus on the problem of nding an upp er b ound on the number of
multiple contacts we brie y reconsider the notion of multiple contact itself What
kind of subspaces of the con guration space are de ned by multiple contacts and
how many obstacles can participate in a multiple contact
The set of placements of the rob ot B in which a certain feature of B is in contact
with a b oundary feature of an obstacle in E of appropriate dimension forms an
f dimensional subspace or hypersurface in the f dimensional con guration
space An intersection of j of these hypersurfaces corresp onds to a simultaneous
contact of the rob ot with j obstacle b oundary features in E Such an intersection
is an f j dimensional subspace of the con guration space Consequently the
f fold contacts app ear at isolated p oints in the con guration space and hence x
the p osition of the rob ot Contacts that involve more than f obstacle features do
not app ear if we assume that the obstacles are in general p osition Such contacts
CHAPTER THE COMPLEXITY OF THE FREE SPACE
can b e discarded without a ecting the complexity of the free space We see that a
rob ot B with f degrees of freedom can have up to f simultaneous contacts with the
b oundaries of the obstacles in E
We consider the situation where a rob ot B moves amidst n k fat obstacles E E
in general p osition where k is a constant The rob ot as well as the individual
obstacles are assumed to have constant complexity so the number of rob ot features
is O and the number of b oundary features in the obstacle set E is O n As
a consequence the total number of hypersurfaces is O n The hypersurfaces are
assumed to b e algebraic of b ounded degree so that the intersection of any j hy
p ersurfaces consists of at most a constant number of connected comp onents To
successfully apply Prop erty the rob ot B is assumed to b e not to o big compared
to the obstacles Let b e a lower b ound on the minimal enclosing hypersphere radii
of all obstacles The reach of the rob ot B is constrained by b where b
B B
is some p ositive constant This assumption regarding the size of the rob ot is not
very restrictive it basically rules out the situation where the rob ot B is so large
that it would make the obstacles into p oint obstacles relative to its own size The
assumption will b e satis ed in most practical cases In the previous subsection we
already saw that such a restriction is required to obtain low free space complexities
We summarize the assumptions b elow
d
The workspace W of the rob ot B is the d dimensional Euclidean space IR
The workspace W of the rob ot B contains a collection E of n k fat obstacles
d
E IR in general p osition for some constant k
The reach of the rob ot B is b ounded by b where b is a
B B
constant and is a lower b ound on the minimal enclosing hypersphere radii
of all obstacles E E
The rob ot B has constant complexity
Each obstacle E E has constant complexity
The hypersurface in the con guration space corresp onding to the set of rob ot
placements in which a certain rob ot feature is in contact with a certain obstacle
feature is algebraic of b ounded degree
The assumptions remain valid throughout the remaining chapters
The proximity result given in Prop erty is the key to successful application
of a pro of strategy that rep eatedly considers an obstacle E and counts the number
of multiple contacts for the rob ot B involving E and obstacles with larger minimal
enclosing hypersphere radii Prop erty guarantees that we nd a constant upp er
b ound on this number for each obstacle E The resulting overall number of multiple
contacts will b e linear which is stated in Theorem
FAT OBSTACLES AND THE FREE SPACE COMPLEXITY
Theorem Let d k and b be constants and let E be a set of n k fat
d
obstacles in IR of constant complexity each and with minimal enclosing hypersphere
radii at least The robot B with constant complexity f degrees of freedom and
d
reach b moves in W IR amidst the obstacles of E Then for each
B
j f the number of j fold contacts of the robot B is linear in the number of
obstacles O n
Pro of Consider some obstacle E E and let b e its minimal enclosing
E
hypersphere radius Let us count the number of j fold contacts of B that involve E
and obstacles E with larger minimal enclosing hypersphere radii Such an obstacle
E must lie within a distance from E in order to allow B to touch E and E
B
simultaneously b ecause the reach of B b ounds B s maximum diameter by
B B
Let p b e the number of obstacles E that lie within a distance from E Since
B
d
b b we know by Prop erty that p k b O
B E
A single j fold contact is determined by j di erent pairs each pair consisting
of a rob ot feature and an obstacle feature Let us assume that the rob ot has x
B
di erent features and that the number of features of each obstacle E is b ounded
by x The rst contact is a contact b etween a rob ot feature and a feature of the
E
obstacle E We have at most x x choices for this contact For each of the j
B E
remaining contacts we can choose the obstacle feature on each of the p obstacles in
the proximity of E which gives a total number of x p p ossibly involved obstacle
E
features For each contact we can again choose from all x rob ot features Hence
B
j
the total number of j fold contacts involving E is b ounded by x x p x x
B E B E
which is a constant
Adding all the n constant upp er b ounds results in an overall upp er b ound on
j
the number of j fold contacts of n x x p x x which is O n since
B E B E
x x p and j are constants
B E
Note that the value of j in Theorem ranges from to f The number of single
contacts is of course also linear b ecause the number of pairs of a rob ot feature and
an obstacle feature is linear The case j is delib erately excluded from Theorem
to emphasize that fatness only reduces the number of multiple contacts and
not the number of single contacts
The f j dimensional subspace de ned by a single j fold contact is not nec
essarily connected Figure shows an example for f and j where it is
imp ossible for the rob ot to move from Z to Z without losing contact with either
!
the upp er or the lower obstacle feature The dimensional subspace induced by
the contact with b oth features is therefore non connected Our assumption that all
contact hypersurfaces are of b ounded degree however implies that the number of
di erent connected subspaces induced by a single multiple contact is b ounded by
some small constant The complexity of the free space is now solely determined by
the number of multiple contacts since the contribution of a single multiple contact
to the free space apparently has constant complexity Variable j in Theorem can
CHAPTER THE COMPLEXITY OF THE FREE SPACE
Z Z
0 1
Figure There is no continuous motion of the rob ot from Z to Z during which
!
it remains in contact with b oth features
only have f di erent values so the total number of multiple contacts is linear
and hence the free space has linear complexity
Corollary Let d k and b be positive constants and let E be a set
d
of n k fat obstacles in IR of constant complexity each and with minimal enclosing
hypersphere radii at least The robot B with constant complexity f degrees of
d
freedom and reach b moves in W IR amidst the obstacles of E Then
B
the free space for the robot B moving amidst the k fat obstacles of set E has linear
complexity
The linear upp er b ound on the free space complexity obviously imp oses an equal
b ound on the complexity of a single free cell
The constant that we obtained in Theorem can b e quite high the rst contact
for the rob ot is a feature of E but each of the other j contacts are chosen from
all features in the proximity of E In practice this approach yields a b ound that
is far from tight b ecause many of the features in E s proximity cannot b e touched
by B while it touches some feature of E Figure shows an example of such a
situation Even though the distance b etween two features alone may allow the rob ot
to touch b oth of them simultaneously the p ositions of the obstacles in the workspace
may prevent the rob ot from actually doing so So only a subset of all theoretical
combinations of features really implies a multiple contact in the con guration space
Clearly the number of actual j fold contacts for B will remain far b elow the upp er
b ound of Theorem
The framework of assumptions that leads to the linear free space complexity
includes a general p osition assumption for the obstacles The assumption basically
simpli es the analysis by allowing us to neglect multiple contacts involving more
than f pairs of features The upp er b ound of f on the value of j in counting
the number of j fold contacts involving a feature of some obstacle E and features
FAT OBSTACLES AND THE FREE SPACE COMPLEXITY
E
e
E
e
E
B
Figure The rob ot B touching the edge e of obstacle E is long enough to touch
e and the edge e of E simultaneously Nevertheless it is unable to do so b ecause
the obstacle E is in its way
of larger obstacles E however seems in no way relevant in obtaining a constant
b ound The general p osition assumption although common in motion planning is
therefore not very essential to the validity of our result
A second glance at the pro of of Theorem learns that most of the assumptions
are not used explicitly Instead the combination of the assumptions leads to a low
obstacle density in the workspace which basically means that any workspace region
with size comparable to the reach of the rob ot intersects only a constant number
of obstacle features This implied workspace prop erty rather than the individual
assumptions is essential to the pro of of Theorem The linear b ounds on the
numbers of j fold contacts can therefore b e extended to motion planning problems
for constant complexity rob ots in workspaces that satisfy the low density prop erty
If in addition the constraint hypersurfaces de ned by the rob ot obstacle contacts
are algebraic of b ounded degree then the linear free space complexity result of
Corollary extends to such motion planning problems as well
! CHAPTER THE COMPLEXITY OF THE FREE SPACE
Chapter
Existing algorithms and fat
obstacles
Exact motion planning algorithms pro cess the free space FP for answering path
nding queries The linear complexity result for FP do es not directly imply that
the outcome of the pro cessing a representation of FP has linear complexity as well
Moreover if an algorithm succeeds in supplying a linear complexity representation
then it may still take far more time to compute this representation
Before we fo cus on a general paradigm for motion planning amidst fat obsta
cles in the next chapter we study the in uence of fatness on a number of existing
algorithms for moving a translating and rotating rigid rob ot among p olygonal ob
stacles This sp eci c problem has b een studied extensively in the mid s which
has resulted in a number of algorithms with varying e ciency In fact planar
motion amidst p olygonal obstacles is the most extensively studied motion planning
problem The algorithms that are discussed b elow constitute an interesting cross
section of the available algorithms and illustrate as such the di erences b etween the
various approaches to solving the problem We consider examples of b oth ma jor
exact approaches to motion planning cell decomp osition and retraction The aim
of studying the algorithms is to learn the sp eci c combinatorial and algorithmic
prop erties that lead to e cient motion planning algorithms so that we can use the
results in nding an e cient paradigm for general fat motion planning
The running time of the b oundary retraction algorithm for a ladder among p oly
gons by Sifrony and Sharir is sensitive to the number K of pairs of obstacle
corners that lie less than the length of the ladder apart Section con rms the in
tuitive feeling that the low obstacle density implied by the fatness and the b ound on
the relative sizes of the rob ot and the obstacles cause K to b e only O n instead of
O n The p erformance of the Voronoi based retraction algorithms though
is not enhanced by our assumptions as the worst case size of the resp ective Voronoi
diagrams do es not b ene t from them
Sections and consider two cell decomp osition algorithms The famous
Piano Movers algorithm for planning the motion of a ladder or p olygon among
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
p olygons by Schwartz and Sharir app ears to b e surprisingly sensitive to the
complexity of the free space The algorithm outputs a decomp osition of the free
space into O n sub cells for problems involving a b ounded size rob ot amidst fat ob
stacles whereas the worst case number of sub cells for general settings can b e as high
as O n Without mo di cations the algorithm computes such a cell decomp osition
!
in time O n A number of adaptations enhance the running time to O n log n
We discuss these results quite thoroughly as they provide the main ideas for the
general paradigm of Chapter The algorithm by Leven and Sharir for a lad
der among p olygons do es not b ene t from the fatness of the obstacles although its
worst case b ehavior is sup erior to that of Schwartz and Sharir We give an example
!
with fat obstacles that leads to a cell decomp osition consisting of n sub cells
which equals the worst case number of sub cells for general obstacles
In Section the claim of Avnaim Boissonnat and Faverjon that their
b oundary cell decomp osition algorithm p erforms considerably b etter than the worst
"
case O n log n running time if the workspace has a low obstacle density is con rmed
for workspaces that satisfy our assumptions We nd that the running time of the
algorithm indeed reduces to O n log n
Throughout the entire chapter it is assumed that a constant complexity rigid
rob ot B moves in a two dimensional Euclidean workspace amidst a collection E of
n p olygonal k fat obstacles for some p ositive constant k Each individual obstacle
E E has constant complexity and a minimal enclosing hypersphere radius of at
least As an exception the b ounds on the sizes of ladder rob ots are expressed in
terms of their lengths instead of their reaches to conform to the original pap ers
Needless to say is that the length of a ladder is closely related to its reach Note
that the rob ot itself need not b e fat The assumptions are generally omitted in the
formulation of the results of this chapter
Boundary vertices retraction
Let us rst consider the b oundary vertices retraction metho d of Sifrony and Sharir
for planning the motion of a ladder amidst p olygonal obstacles which runs in
time O K log n where K is the number of pairs of obstacle corners vertices that
lie less than the length of the ladder apart We will prove that in the indicated
setting K O n yielding an e cient O n log n algorithm
For simplicity we assume that the p olygonal obstacles in E E E are
# n
ordered by increasing minimal enclosing circle radii and furthermore that
# n
the features of each ob ject E are ordered in some way f f c O by the
i i# ic
constant complexity of E Note that a feature that app ears after feature f in the
i
lexicographical ordering of features either b elongs to the same obstacle as f or to a
larger obstacle
Lemma b ounds the number of feature pairs with small mutual distance If
we charge each such close pair to the lexicographically smallest of the two involved
FATNESS SENSITIVE CELL DECOMPOSITION
features then rst of all each pair is counted but more imp ortantly it turns out
that each feature gets charged only a constant number of times As a result the
total number of chargings and hence close feature pairs adds up to O n
Lemma Let k and b be positive constants and let E be a set of polygonal k fat
constant complexity obstacles with minimal enclosing circle radii at least Then
the number of feature pairs edges corners that lie less than b apart is O n
Pro of Let us count all lexicographically larger features f that lie within a distance
b from a feature f Clearly the distance from the obstacle E containing f to
i
the feature f is b ounded by b b The lexicographically larger feature f
i
b elongs by de nition either to E or to an obstacle E with j i By Corollary
i j
the number of such obstacles E within a distance b from E is b ounded by
j i i
a constant Combined with the constant complexity of these obstacles and of E it
i
follows that there is at most a constant number of choices for f The O n b ound
follows after summing over all f
Lemma proves that K O n We obtain the following nal result
Theorem Sifrony and Sharir s boundary vertices retraction algorithm
plans the motion of a ladder robot B with length b amidst the fat obsta
cles of E in time O n log n for any constant b
Fatness sensitive cell decomp osition
This section considers the combinatorial and algorithmic consequences of fatness for
the famous Piano Movers algorithm by Schwartz and Sharir Subsection
shows that the complexity of the cell decomp osition of FP computed by the metho d
is O n under the assumption of fat obstacles and a b ounded size rob ot whereas
the b ound is O n in the general case The algorithmic part in Subsection
deals with the e cient computation of the decomp osition the subsection improves
!
the direct b ound of O n to O n log n
Schwartz and Sharir apply the cell decomp osition technique to obtain an
O n algorithm for planning the motion of a ladder B moving amidst p olygonal ob
!
stacles E in the plane Their metho d decomp oses W IR into so called noncritical
!
regions lifts these regions into three dimensional cylinders in C IR
decomp oses the free part of the cylinders into sub cells and nally captures the ad
jacency of the sub cells in a connectivity graph We will go into more detail on each
of these steps and while doing so fo cus on the consequences of fatness for each of
these steps We emphasize that we will not give an extensive explanation of the
ladder algorithm The reader is referred to the original pap er or Latombe s
b o ok for a detailed description
!
The noncritical regions in the rob ot s workspace W IR are de ned by critical
curves The meaning of these curves is not imp ortant in our analysis so we restrict
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
ourselves to summarizing the di erent types of critical curves We adopt the clas
si cation of the critical curves used in Latombe s b o ok Let P and Q b e the
endp oints of the ladder B and let jP Q j b e its length Cho ose P as the rob ot s
reference p oint
An obstacle edge is a critical curve of type
Let e b e an obstacle edge The line segment at a distance from e is a critical
curve of type The length of the critical curve equals the length of e
Let x b e an obstacle corner and let e and e b e the edges emerging from x
!
The circular arc with radius centered at x and running b etween the half
lines starting at x and containing the edges e and e resp ectively is a critical
!
curve of type
Let x b e a convex obstacle corner and let e b e one of the edges emerging from
x The line segment traced out by P while B slides along e so that Q touches
e and x touches B is a critical curve of type The curve is the extension
with length of the edge e at x
Let x and x b e convex obstacle corners such that the line passing through
!
x and x is tangent to the obstacle set E in b oth x and x The line segment
! !
traced out by endp oint P while B slides along x and x is a critical curve of
!
type Note that the distance from x to x must b e less than
!
Let x b e a convex obstacle corner and let e b e an obstacle edge such that x is
not an endp oint of e The curve traced out by P while Q slides along e and
while B remains in contact with x is a fourth degree critical curve of type
Note that again the distance from x to e must b e less than
Figure illustrates the various critical curve types The intersecting critical
!
curves partition W IR The part of a critical curve b etween two p oints of inter
section with other critical curves is called a critical curve section A p osition x y
of the rob ot B is admissible if there exists an orientation such that x y FP
A noncritical region is a maximal subset of admissible rob ot p ositions intersecting
no critical curves Hence the critical curves determine a set of noncritical regions
in W
Let f j x y FP g b e the set of free orientations of B with P xed
x y
at a p oint x y in a noncritical region R The set consists of a nite number of
x y
op en maximum connected intervals For each such interval b oth the
! x y
rob ot placement x y and the rob ot placement x y are placements in which
!
the rob ot touches the obstacle set The unique stop touched by B in the contact
placement x y resp x y is denoted by s x y resp s x y The
! !
set of all pairs s x y s x y such that is referred to as x y
! ! x y
The critical curves are de ned so that for each pair of p oints x y and x y in a
FATNESS SENSITIVE CELL DECOMPOSITION
e
!
e
x
e
e
type
type type
x
x
e
x
x
!
e
type type
type
Figure The six types of critical curves in Schwartz and Sharir s solution to the
Piano Movers problem
single noncritical region R the sets x y and x y are equal In the sequel
we use the abbreviation R x y where x y is any p oint in R Each pair
of contact p ositions s s R de nes a cell in the cell decomp osition of FP
!
Schwartz and Sharir s metho d rst computes all critical curves and then all
intersections of the curves resulting in a collection of intersection p oints and a
collection of critical curve sections With each intersection p oint we store the critical
curve sections that are incident at this intersection p oint Each critical curve section
separates two regions we arbitrarily call one of the regions left and the other
one right Next we compute left and right de ne a connectivity
graph no de for each sub cell s s induced by the regions left and right and
!
build the adjacency relation based on the adjacency of the corresp onding sub cells
b etween the no des induced by b oth regions Note that each no de is generated by
a single critical curve section If we rep eat this pro cedure for every critical curve
section each sub cell is represented in the connectivity graph as many times as there
are critical curve sections b ordering the region that induced the sub cell All no des
that corresp ond to the same sub cell are circularly connected
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
Complexity of the cell decomp osition
We recall that the number of obstacle edges and corners is O n First we observe
that the number of type curves is not in uenced by the fatness of the obstacles
and is O n in b oth the arbitrary and the fat case
In the general case of arbitrary p olygonal obstacles the number of type curves
is O n since each pair of corners may de ne such a curve The same number
applies to type curves since each pair of one edge and one convex corner may
induce such a curve As a result the total number of critical curves is O n Each
!
pair of curves may intersect so that there can b e up to O n intersections and
!
hence O n critical curve sections
Things change if we assume the obstacles to b e k fat Let us rst observe an
imp ortant prop erty of all critical curves following from the de nitions of the curves
Prop erty Each point on a critical curve is less than the length of the ladder
away from the obstacle features corners edges that de ne this curve
The other imp ortant to ols in the analysis of the fat case are the low ob ject density
results Theorem Corollary and Lemma Like in the previous section
the length of the rob ot B P Q is b ounded by b b for all i n
i
Lemma b ounds the number of critical curves of type and The lemma
follows more or less directly from the de nition of the curves and Lemma which
b ounds the overall number of feature pairs and hence the number of feature pairs
de ning critical curves of type and
Lemma The number of critical curves of type and in the workspace is O n
Pro of Each pair of obstacle corners with mutual distance at most may under
some additional conditions like relative angles of incoming edges de ne one critical
curve of type The number of corner pairs lying less than apart is b ounded
by O n by Lemma Thus the number of type curves is b ounded by O n
Each pair of an obstacle corner and an obstacle edge with mutual distance at most
may again under some additional conditions de ne one curve of type Lemma
b ounds the number of such pairs and hence the number of type curves by
O n
Each of the of O n critical curves may b e intersected by other critical curves and
as a result of that b e cut into a number of critical curve section Since each critical
curve is de ned by one or two obstacle features an intersection p oint p
can b e regarded as b eing implied by the union of the two sets of de ning features
Each intersection is as such implied by a collection of two three or four obstacle
features If we now charge each intersection p to the lexicographically smallest of
these features then we nd again that each feature is charged at most a constant
number of times Adding up all contributions of features f leads to a total of O n
intersections and hence critical curve sections
FATNESS SENSITIVE CELL DECOMPOSITION
Lemma The number of critical curve sections in the workspace is O n
Pro of Let us b ound the number of critical curve intersections p implied by f and
a set F of lexicographically larger features jF j the features in ff g F
de ne the two curves and intersecting in p By Prop erty the distances from
the intersection p oint p to the de ning features ff g F of and do not exceed
As a result the distance from any feature f F to f is at most Moreover
the distance from the ob ject E containing f to any feature f F is b ounded by
i
b b Each lexicographically larger feature f F b elongs either to
i
E or to an obstacle E with j i By Corollary the number of such obstacles
i j
E within a distance b from E is b ounded by a constant Combined with the
j i i
constant complexity of these obstacles and of E we nd that there is also at most a
i
constant number of candidates for inclusion in F Hence there exist only a constant
number of sets F of features that together with f can imply a pair of intersecting
curves and The low degree of the curves implies that the number of intersec
tions of a pair of curves and is b ounded by a constant so any choice for F
can contribute no more than O intersections Adding up the contributions of all
features f yields a total of O n intersections and hence critical curve sections
We have seen that the number of connectivity graph no des added by the critical
curve section equals the number of sub cells induced by the region left plus the
number of sub cells induced by the region right If is a type curve only one
of the two regions is noncritical in all other cases b oth regions will b e noncritical
A region that is not noncritical will induce no graph no des
Now we analyze the number of sub cells induced by a single noncritical region
R In Schwartz and Sharir s metho d each pair s s R de nes a sub cell
!
Hence the number of sub cells induced by a noncritical region R is determined by
the number of pairs in R each pair in R consisting of two di erent contact
placements for B with P xed at some p oint in R The number of sub cells induced
by R is therefore determined by the number of di erent contact placements for B
when we x its endp oint P at some p oint x y in R and vary its orientation
Note that due to the shap e of the features each feature can b e touched by B in
at most two di erent orientations while its endp oint P is xed at x y In the
case of arbitrary p olygonal obstacles we can easily construct examples where the
rob ot can touch any of the O n obstacle features each at a di erent orientation
A noncritical region R can therefore induce O n sub cells Since the number of
" #
noncritical regions is O n we obtain a total number of O n sub cells As b efore
things are di erent in a fat setting It is easy to see that an obstacle feature f
touched by B with P xed at x y in some contact p osition must lie close to x y
Using the low density prop erty of spaces with fat ob jects we can b ound the number
of such features f
Lemma Each noncritical region in the workspace induces only O subcells in
the con guration space
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
Pro of The number of sub cells induced by a noncritical region R dep ends on the
number of obstacle features that can b e touched by B P Q while its endp oint P
is xed at some arbitrary p oint x y R Obviously such a feature intersects the
circular region S obtained by rotating B while keeping P at x y The number of
ob jects E i intersecting the circle S with radius b is constant by Theo
i
rem As a consequence the number of features f intersecting S and p otentially
touched by B is O So the number of sub cells induced by a noncritical region
R is b ounded by a constant
Lemma shows that each critical curve section adds at most twice a constant
number of no des to the connectivity graph By Lemma we conclude that the
total number of no des is O n Let us count the adjacencies of a single no de N
added by some section Assume without loss of generality that N is induced by
the noncritical region left The no des that are adjacent to N either corresp ond to
the same sub cell or are induced by the noncritical region right and added by the
section As all no des corresp onding to a single sub cell are circularly connected the
number of adjacencies of the rst type cannot exceed two The number of adjacent
no des of the second type is constant by Lemma Hence each sub cell is adjacent
to a constant number of other sub cells resulting in a total of O n graph edges
Theorem The connectivity graph corresponding to the cell decomposition of the
free space of a ladder moving amidst k fat obstacles has O n nodes and edges
Computing the cell decomp osition
Although the complexity of the connectivity graph corresp onding to the cell de
comp osition is O n a straightforward application of Schwartz and Sharir s metho d
would result in O n time to compute the decomp osition This b ound turns up
in each of the three steps in the algorithm the rst step where all critical curves
are computed the second step where all critical curve inter sections are computed
and the third step where all sub cells induced by a single noncritical region are de
termined If we incorp orate the plane sweep ideas by Bentley and Woo d for
rep orting geometric intersections in each of the three steps then the e ciency of
each individual step is enhanced to O n log n For a discussion of the main in
gredients of a plane sweep we refer to Section Here we con ne ourselves to
mentioning that the K intersections of n line segments in the plane can b e rep orted
in time O n K log n The ideas are straightforwardly generalized to x monotone
constant complexity curves Below we rede ne each of the three ab ove steps as a
problem of rep orting constant complexity curve intersections
We have shown in the previous subsection that the number of type or curves
is linear in the case of fat obstacles Each of these curves is determined by two
features We could naively try all p ossible pairs of features to nd out which pairs
generate a curve This strategy would require n time to nd the O n curves
FATNESS SENSITIVE CELL DECOMPOSITION
Instead we should use the knowledge that only two features that lie less than apart
can de ne a curve of type or Let us rst determine all pairs of features that lie
less than apart By Lemma there exist O n such pairs After determining
the close feature pairs O n time su ces to nd which of the corner corner and
corner edge pairs indeed de ne curves of types and
To determine the pairs of features with distance at most we use the ideas
of Sifrony and Sharir which in turn rely on the techniques from Sifrony
and Sharir wrap each obstacle edge and its two endp oints by a so called envelope
The envelope of an edge e is the set e S the b oundary of the Minkowski
O
di erence of e and the circle with radius centered at the origin Hence the
envelope of an edge e equals the set of p oints with distance to e As such it
consists of two straight segments parallel to e and two circular arcs of arc length
half circles ab out e s endp oints see Figure It is not to o hard to see that
e
Figure The envelope of an edge e and its two endp oints
intersect if and only if features v the envelopes of the edges e v v and e v
!
!
g lie less than apart As a result we can nd the v from fe v v g and fe v
! !
!
close pairs of features by sweeping the constant complexity envelope curves in the
plane To satisfy the input requirement that the curves are x monotone we simply
cut the circular arcs into two subarcs at x extremal p oints The envelope curves are
!
lab eled with the corresp onding edge and endp oints
Let us now consider the e ciency of the ab ove sweep for envelope intersections
and hence for close feature pairs By the convexity of the envelopes two envelopes
can only intersect in a constant number of p oints Moreover each rep orted inter
section of two envelopes leads to a non zero number of close feature pairs As a
result the number of envelope intersections is of the same order of magnitude as the
number of close feature pairs O n by Lemma Rep orting the K O n enve
lop e curve intersections with the plane sweep takes O n K log n O n log n
time From the envelope intersections the close feature pairs and subsequently the
critical curves of type and can b e computed in O n time
!
Note that envelopes of consecutive edges on the b oundary of a single obstacle may partially
coincide Potential problems with these coinciding parts are avoided if we merge these parts into a single curve lab eled with all corresp onding edges and endp oints
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
After having computed the critical curves in O n log n time we encounter a
similar problem when computing the curve intersections and the resulting critical
curve sections We could naively sp end O n by intersecting each curve with
any other curve although we know that the number of intersection is only O n by
Lemma An obviously b etter idea is to cut the constant complexity critical
curves into a constant number of x monotone sub curves and feed these to the plane
sweep The sweep rep orts the K O n intersections in time O n K log n
O n log n The intersection p oints cut the critical curves into the desired critical
curve sections
For each of the O n critical curve sections we have to compute left and
right Computing left requires choosing a p oint x y left xing
the rob ot s endp oint P at x y and rep orting all features that can b e touched by
B and the orientations in which they are touched which seems rather di cult to do
e ciently Fortunately the set of features that are to b e rep orted forms a subset
of the set of features of the O obstacles Theorem that intersect the circular
region S centered at x y and with radius Checking all O n obstacles for
x y
intersection with S is surely not the most e cient way of determining the O
x y
obstacles that intersect the circle A b etter idea is to use the results on b ounded
size range searching from Chapter The construction of the data structure storing
all obstacles of E takes O n log n log log n time The O n queries with circles S
with radius induced by all critical curve section then take O n log n in total
The entire computation takes O n log n log log n The following two step approach
however avoids the log log n factor rst nd for all circles S the set V S of
#
obstacles that have a vertex inside S and then rep ort for all circles S the set V S
of obstacles E with edges that intersect the b oundary S of S Notice that the
union of p ossibly overlapping sets V S and V S is clearly the set of all obstacles
#
intersecting S Below we solve the subproblems one by one
The computation of the constant cardinality sets V S can b e further simpli
#
ed by realizing that V S is a subset of the set of obstacles having a vertex inside
#
the axis parallel minimal enclosing square C of S The enclosing square C of the
circle S with radius has side length b By Theorem the number of
obstacles from E intersecting C is constant so de nitely the number of obstacles
having a vertex inside C is constant To solve the reduced problem we store all ob
stacle vertices in a data structure that supp orts e cient axis parallel or orthogonal
range searching queries Preparata and Shamos text b o ok on computational
geometry gives an appropriate data structure based on the layered range tree The
characteristics of the layered range tree structure are given as Lemma
Lemma There exists a data structure of size O n log n that answers planar
range search queries among points in time O log n K where K is the number of
answers to the query Building the structure requires O n log n time
After having sp ent O n log n time to build the range searching structure we can
easily rep ort all sets V S in time O n log n by querying the structure with the #
FATNESS SENSITIVE CELL DECOMPOSITION
enclosing squares and subsequently ltering out the obstacles that do not have a
vertex inside S
The nature of the second subproblem rep orting intersections of circles and
obstacle edges suggests the use of the plane sweep for rep orting intersections of
constant complexity curves By the pro of of Lemma or Theorem each circle
S S is intersected by only a constant number of edges This observation
x y
not only implies that each set V S has constant cardinality but also that the total
"
number of circle edge intersections is O n Unfortunately we do not only encounter
circle edge intersections throughout the sweep the circles also intersect each other
These intersections are non interesting events with resp ect to nding the sub cells
induced by the non critical regions but they do a ect the e ciency of the plane
sweep The number of such irrelevant circle circle intersections seems imp ossible
to b ound without further provisions
Before analyzing the number of circles intersecting a given circle we recall that
each critical curve section de nes two circles centered in left and right
resp ectively If we allow these centers to b e anywhere in left and right then
it seems imp ossible to b ound the number of circle circle intersections However if
the centers of b oth circles are restricted to the vicinity of their implying critical
curve sections then proving an O n b ound on the number of intersections b ecomes
feasible In the mo di ed version of Schwartz and Sharir s algorithm we therefore
take care to choose the circle centers within a distance say from the originating
critical curve section thus allowing for an e cient sweep of the arrangement of
curve and circle parts
Lemma If the centers of the circles implied by the O n critical curve sections
are chosen within a distance from then the number of circle circle intersection
is O n
Pro of Let us consider a circle S and count the number of circles intersecting
m
S Any circle S intersecting S must clearly have its center m lying in the
m m m
circle S As m was chosen within a distance from the critical curve section
m "
de ning S this de ning section must intersect the again larger circle S
m m #
By Prop erty in turn any p oint on lies within a distance from all its de n
ing features which must therefore intersect the circle S By Theorem the
m $
number of obstacles and hence obstacle features intersecting S is b ounded by
m $
a constant This constant number of features can de ne at most a constant number
of critical curves that may p ossibly intersect S The constant size subset of
m #
critical curves that intersect S de ne only a constant number of critical curve sec
m #
tions As each of the O sections de nes two circles the total number of circles
de ned by such curve sections is constant The circles S that intersect S must
m m
b elong to this constant size set of circles Hence any circle S is intersected by
m
O other circles leading to the O n b ound on the total number of intersections
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
The ab ove lemma shows that the total number K of intersections encountered
during the sweep of the arrangement of the O n prop erly chosen circles and the
O n non intersecting obstacle edges equals O n The running time of the sweep
therefore amounts to O n K log n O n log n The output of the sweep are
the constant size sets V S of obstacles E E From the constant size unions
V S V S the sets R with R m can b e computed in constant time
! m m
The mo di ed version of Schwartz and Sharir s algorithm which is tailored to
fat obstacles now consists of three steps the rst two are plane sweeps and the
third combines a sequence of range search queries with a plane sweep Each of the
three steps runs in time O n log n so that the running time of the entire algorithm
amounts to O n log n as well The result is summarized in the following theorem
Theorem Schwartz and Sharir s cell decomposition algorithm can be
adapted to plan the motion of a ladder robot B with length b amidst the
fat obstacles of E in time O n log n for any constant b
A p olygonal rob ot
In the preceding two subsections we have restricted our attention to a ladder moving
amidst p olygonal obstacles Schwartz and Sharir s pap er however also gives
an algorithm for a p olygonal rob ot
The algorithm for a p olygonal rob ot is similar to the algorithm for a ladder
The only di erence concerns the de nition of the critical curves There are more
and di erent types of critical curves in the p olygonal case Although the critical
curves are di erent the basic prop erties of these curves remain valid features that
are involved in the de nition of a single critical curve are less than the diameter of
the rob ot apart and each p oint on a critical curve is less than the diameter of the
rob ot away from the features that de ne it The validity of these prop erties allows
us to use a similar pro of strategy and a similar approach for an algorithm in the
case of a p olygonal rob ot resulting in the same O n log n complexity for the cell
decomp osition and for the motion planning algorithm
A fatness insensitive cell decomp osition
A di erent example of the cell decomp osition approach is the algorithm of Leven and
Sharir presented in which also applies to a ladder moving in a two dimensional
workspace amidst p olygonal obstacles Although the worst case cell decomp osition
size and running time of the algorithm for general obstacles O n and O n log n
"
resp ectively are sup erior to the O n worst case b ounds for the Schwartz Sharir
algorithm a simple example shows that the Leven Sharir algorithm is inferior when
the obstacles in the workspace are fat More precisely the example shows that
motion planning problems with fat obstacles can still give rise to a decomp osition
of the free space into n sub cells
A FATNESS INSENSITIVE CELL DECOMPOSITION
The basis of the algorithm by Leven and Sharir is the fact that a simple O n
cell decomp osition exists for the strictly translational version of the problem This
cell decomp osition is o course a decomp osition of the pro jective subspace IR of
the con guration space IR of the original problem The decomp osition is
such that a small change in the orientation of the rob ot leads to an only slightly
di erent and often top ologically equivalent cell decomp osition In con guration
space we conceptually obtain an in nitely small grain stack of such continuously
varying planar cell decomp ositions If one would descend the stack then at certain
orientations the planar cell decomp osition changes top ologically These so called
critical orientations divide the angular dimension of the con guration space
into intervals of similar planar cell decomp ositions The one dimensional intervals
I de ne slices IR I in con guration space that cut the stack into sub
stacks Corresp onding regions in di erent layers of the stack form a sub cell in the
decomp osition of the con guration space All three dimensional sub cells in a slice
span the entire slice from its lower b oundary to its upp er b oundary
! "
Sub cells app ear or disapp ear only at slice b oundaries Moreover assuming general
p osition of the obstacles exactly one sub cell app ears or disapp ears at any slice
b oundary
The complexity of the resulting cell decomp osition is determined by the number
of critical orientations which form the interval endp oints and hence the slice
b oundaries A critical orientation is an orientation for which one of the three
conditions listed b elow is true is the length of the ladder The identi cation of
the conditions is adopted from the pap er by Leven and Sharir
C There exist two obstacle corners such that the op en line segment connecting
them is entirely contained in the closure of W n E and has orientation
E E
C There exist an obstacle corner and a p oint on some obstacle edge such that the
op en line segment connecting the corner and the p oint is entirely contained in
W n E and has orientation and length
E E
C There exist two p oints on two obstacle edges and an obstacle corner c such
that the op en line segment connecting the two p oints has orientation and
length passes through c and is entirely contained in W n E except at
E E
c
Clearly the number of critical orientations in the case of arbitrary p olygonal ob
stacles is O n The resulting free space decomp osition consists of O n cells In
contrast to the algorithm in Section the number of critical events is not in
uenced by the p ossible fatness of the obstacles This is most easily seen from a
situation where we have n square fat obstacles placed in circular fashion see
Figure In this example the obstacle corner v can b e connected to any of the
n obstacle corners facing the interior of the circle by a line segment that is
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
v
Figure The number of critical orientations is not reduced by the fatness of the
obstacles
wholly contained in the closure of W n E If we do the same for each of the
E E
remaining n obstacle corners facing the interior of the circle we obtain a total
n n n n di erent line segments each of them number of
!
corresp onding to an o ccurrence of condition C By small p erturbations of the
obstacles we can establish that each of the line segments has a di erent orientation
which results in at least n n critical orientations induced by o ccurrences
!
of condition C As a consequence the number of critical orientations is O n
Thus the fatness of the obstacles do es not lead to a reduction of the complexity of
the cell decomp osition in this case
Boundary cell decomp osition
Avnaim Boissonnat and Faverjon describ e a variant of the cell decomp osition
approach that rather than decomp osing the free space itself decomp oses the free
space b oundary BFP cl FP n FP into simple sub cells or faces The results in
Chapter imply that the complexity of BFP in our realistic setting is O n The
algorithm requires that the complement W n E of the obstacles is b ounded
E E
This requirement is easily met by enclosing the workspace obstacles in arbitrarily
large b ox After the decomp osition of the free space b oundary additional faces are created to establish su cient connectivity among the faces to solve the path nding
BOUNDARY CELL DECOMPOSITION
problem b etween two sp eci c placements The initial and nal placements determine
the faces that are added in this step The preceding decomp osition of BFP on the
other hand is indep endent of the query The b oundary faces and the additional
faces constitute the no des of a graph in which two no des are connected if their
corresp onding faces share a common b oundary The graph is subsequently searched
for a path connecting the initial and nal rob ot placements The motion planning
algorithm has worst case running time O n log n but the authors claim that the
running time improves to O n log n in workspaces of b ounded lo cal complexity
Below we see that the running time is O n log n in the case of a p olygonal rob ot
B with reach b and hence diameter at most moving among the k fat
B B
p olygonal obstacles of E
The free space b oundary decomp osition is based on ideas b orrowed from a pap er
by Avnaim and Boissonnat The authors decomp ose BFP into faces b ounded
by two straight edges parallel to the plane and by two curved arcs They
essentially compute the O n contact surfaces consisting either of all placements in
which a rob ot vertex v touches an obstacle edge e or of all placements in which a
rob ot edge e touches an obstacle vertex v and subsequently subtract the collection
of placements in which a rob ot edge intersects an obstacle edge from each of these
contact surfaces A sweep of each contact surface computes the di erence of the ini
tial surface and the collection of placements corresp onding to intersecting rob ot and
!
obstacle edges The sweep simultaneously sub divides the resulting set di erence
into faces b ounded by two arcs and two straight edges A sweep of a single contact
!
surface takes worst case O n log n time resulting in a total time of O n log n
time for handling all surfaces The faces form a decomp osition of the free space
b oundary into simple sub cells Determining the connectivity of the faces takes time
prop ortional to the cumulative complexity of the faces provided that the b ounding
curves of the faces are lab eled with a characterization of the double contact that
they represent
Let us now consider the consequences of fatness on the decomp osition sketched
ab ove Assume that f is the constant complexity surface consisting of the place
!
ments in which the rob ot feature touches the obstacle feature The ob jective
is to subtract from f all placements Z in which an edge e of B intersects some
!
obstacle edge e Such an obstacle edge e must lie within a distance b
B
from the obstacle feature b ecause B simultaneously touches and intersects e
By Lemma the overall number of such pairs is O n and they can b e computed
in time O n log n using the technique by Sifrony and Sharir outlined in Subsection
Provided that these O n close feature pairs are computed in advance it is
p ossible to determine the m edges e within a distance from in time O m We
B
charge each edge e to the close pair e The m edges e and the O rob ot edges
e de ne O m constant complexity collections of p oints that are to b e subtracted
from f The computation of the di erence of f and these O m sets and the
! !
A suitable parametrization of the contact surface facilitates an e cient sweep
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
simultaneous sub division of the di erence via the surface sweep takes O m log m
time Rep eating the arguments for all contact surfaces f leads to at most two charg
P
ings of every close pair e so it follows that m O n The cumulative
f
f
running time of all sweeps and hence of the entire computation of all faces of BFP
P
equals O m log m O n log n Extraction of the adjacency information for
f f
f
the faces takes O n log n time The resulting graph is denoted by G
B
Assume for the second part of Avnaim Boissonnat and Faverjon s algorithm
that the goal is to nd a path b etween the free placements Z x y and
Z x y Let K and K b e the free cells containing Z and Z resp ectively
! ! ! ! ! !
The b oundedness of W n E implies that all free cells are b ounded as well
E E
The b oundaries K and K may well b e non connected The b oundedness of
!
the cells guarantees that one connected comp onent of a cell b oundary the so called
external b oundary encloses all other connected comp onents Let K and K b e
!
the external b oundaries of K and K resp ectively Furthermore let K b e the
!
intersection of the plane and the free cell K and let K b e the intersection
!
by a of and K The placement Z is connected to any placement in K
! !
semi free path that is entirely contained in K K Similarly Z is connected
!
by a path in K K The key observation is that to any placement Z K
!
!
!
Z and Z are connected by a semi free path if and only if b oth placements b elong
!
to the same free cell K K In that case Z and Z are connected by a path in
! !
K K K K K K A decomp osition of K and K into simple
!
! ! !
faces facilitates path nding in these two subsets of FP
The cross section K of the free cell K is a p olygonal region The complexity
of the p olygonal region K is b ounded by the complexity of the intersection of
the FP and the plane which is O n by the results from Chapter A
vertical decomp osition sub divides K into O n faces b ounded by at most four
edges The computation of the decomp osition and the adjacencies of the faces via
a sweep of K takes O n log n time For details on such a sweep the reader is
referred to Section Let G b e the adjacency graph on the faces in the vertical
decomp osition A similar treatment of K results in an O n decomp osition of K
! !
and a corresp onding graph G The faces containing Z and Z are easily determined
! !
during the sweeps Notice that no face corresp onding to a no de in G is adjacent to
a face corresp onding to a no de in G unless
! !
The nal task is to merge the graphs G G and G into a single graph G on the
B !
faces in all three decomp ositions Merging the graph G into G requires a single
B
simultaneous scan of the no des in G corresp onding to faces that are intersected
B
by and the no des in G corresp onding to faces on the b oundary of K A
no de in G and a no de in G are connected by an edge if the corresp onding faces
B
share a curve of non zero length A careful implementation of the merge requires
O n time Finally the graph G is merged into G G using the same ideas and
! B
hence within the same time b ound
A search of the graph G with size O n returns a sequence of faces connecting
the face containing Z and the face containing Z if and only if Z and Z lie in the
! !
TOWARDS A GENERAL METHOD
same free cell The pap er by Avnaim Boissonnat and Faverjon includes clues
on transforming the sequence of faces into an actual semi free path for the rob ot B
Notice that contrary to most other exact algorithms part of the work is dedicated
to the sp eci c query with the p oints Z x y and Z x y another
! ! ! !
query requires re doing the second part of the construction
The most exp ensive step from a computational p oint of view of the algorithm
sketched ab ove is the computation of the close feature pairs taking O n log n time
The following theorem summarizes the result obtained in this section
Theorem Avnaim Boissonnat and Faverjon s boundary cell decomposition
algorithm can be adapted to plan the motion of a polygonal robot B with diameter
b amidst the fat obstacles of E in time O n log n for any constant b
Towards a general metho d
The preceding sections include a variety of algorithms for the solution of the planar
motion planning problem amidst fat obstacles all running in O n log n time The
main goal of the nal chapters of this thesis however is to nd a more general
solution to the motion planning problem amidst fat obstacles Unfortunately the
algorithms presented here are dedicated to planar problems
Algorithms for e cient motion planning in three dimensional workspaces are
scarce Approaches in contact space like the algorithms by Sifrony and Sharir in
Section and by Avnaim Boissonnat and Faverjon in Section were never
shown to generalize to D workspaces The problem in generalizing such metho ds
lies in the di culty of establishing su cient connectivity among the no des corre
sp onding to vertices or faces in a single free cell to guarantee the exact solution of
the planning problem
The general approaches to motion planning for rob ots with f degrees of freedom
are the cell decomp osition metho d by Schwartz and Sharir running in time
f !
f "
and the roadmap metho d by Canny running in O n log n time The O n
general and recursive nature of these approaches makes it unlikely that they take
advantage of any sp ecial structure of FP if present like in our framework The
rep eated pro jection of the free space in the rst algorithm is likely to destroy any
structure of the free space and may lead to high complexities of the free space
pro jection regardless of the complexity of the original free space The number of
sub cells in the resulting cylindrical decomp osition can therefore b e high despite
a p ossible low complexity of FP The recursive manner of introducing curves to
guarantee the connectivity of the roadmap in Canny s metho d seems insensitive to
a sp ecial structure or low complexity of the free space The number of such curves
relates to some extent to the number of lo cal extrema of the free space and in certain
lower dimensional subspaces of the free space and the fatness of the obstacles in
the workspace do es not seem to reduce the latter quantity
CHAPTER EXISTING ALGORITHMS AND FAT OBSTACLES
The Piano Movers algorithm outlined in Section deviates from the general
f !
O n cell decomp osition approach in that it takes a decomp osition of the two
dimensional subspace W IR of C IR as the basis for a cylindrical
decomp osition while the general approach would take a decomp osition of a one
dimensional subspace of C as a starting p oint The alternative of a cylindrical
decomp osition based in a higher dimensional subspace B of C o ers the opp ortunity
to use prop erties of the space B to obtain e cient decomp ositions The pro jections
of B in the general approach a ect such b ene cial prop erties so that they no longer
hold in the pro jective subspaces of B
A closer lo ok at the details of Section learns that the O n workspace regions
R de ned by the critical curves are such that the intersection of the free space with
their liftings R has constant complexity This prop erty led to Lemma
stating that each region R induces only O sub cells in the free space decomp osition
and hence only O n sub cells in the entire decomp osition of the free space A
recursive decomp osition of W into similar regions R could easily lead to n
regions and thus to n sub cells The ability to de ne a decomp osition like the
one in the rst sentences of this paragraph is ro oted in the relative low obstacle
density in the workspace While the rob ot s reference p oint is con ned to some
su ciently small region R ! W the rob ot is able to touch only a constant number
of obstacle features This fact causes the free part of the con guration space cylinder
R to have constant complexity
The validity of the low obstacle density prop erty for workspaces of arbitrary
dimensions suggests that the ideas in the preceding paragraph are extendible to other
motion planning problems Chapter formalizes and exploits the ideas to obtain a
strategy for motion planning that reduces the problem of partitioning the free space
to the intuitively simpler problem of computing some constrained decomp osition
of the lower dimensional workspace provided that the workspace is a pro jective
subspace of the con guration space The e ciency of the approach dep ends on the
availability of small constrained workspace decomp ositions which is the topic of
Chapter By demonstrating the existence of small decomp ositions the chapter veri es the validity of the approach
Chapter
A paradigm for motion planning
amidst fat obstacles
The aim of this chapter is to determine a general approach to planning the motion of
a not to o large constant complexity rob ot moving amidst k fat constant complexity
obstacles In Section we have seen that the existing planar motion planning
algorithms are not easily extendible towards other problems Moreover the existing
general approaches to motion planning like those by Schwartz and Sharir and
Canny are computationally exp ensive even for problems involving fat ob jects
Motion planning problems in Euclidean workspaces of dimension three normally
imply at least three dimensional con guration spaces A con guration space con
tains constraint hypersurfaces of the form f consisting of placements of the rob ot
B in which a rob ot feature is in contact with an obstacle feature We shall denote
the fact that is a feature of some ob ject or ob ject set X by X The arrange
f
ment of all constant complexity constraint hypersurfaces f B E
f f
divides the higher dimensional con guration space into free cells and forbidden cells
Even in the case of fat motion planning the complexity of a single free cell can b e
O n which illustrates that some additional pro cessing is necessary to facilitate
e cient motion planning Naturally the structure of a higher dimensional arrange
ment like the arrangement of constraint hypersurfaces is complex to understand let
alone to sub divide the free arrangement cells into simple sub cells or to catch their
structure in some one dimensional roadmap At this p oint however fatness comes
to our help to provide us with a very useful prop erty of an f dimensional con gura
tion space C of the form C W D where W is the d dimensional workspace and
D is some f d dimensional rest space Free ying rigid rob ots for example
!
t well in this framework For a free ying rigid rob ot in W IR D is the space
de ned by the three rotational degrees of freedom of the rob ot The low ob ject
density prop erty of the workspace implied by the fatness of the obstacles can b e
shown to result in a very interesting prop erty of con guration space namely that
CHAPTER A PARADIGM FOR FAT MOTION PLANNING
for each p oint p W
jf f j B E f p D gj O
f f
In words the f d dimensional subspace p D obtained by lifting the workspace
p oint p into con guration space is intersected by only a constant number of con
straint hypersurfaces An immediate consequence of this result is that the hyper
surfaces de ne a constant complexity arrangement in each cross section p D of the
con guration space C
At a more abstract level motion planning problems for free ying rob ots amidst
fat obstacles can b e regarded as a sub class of the larger class of motion planning
problems with con guration spaces C B D that satisfy for each p oint p B
jf f j B E f p D gj O
f f
In general a con guration space C that satis es this constraint will b e said to
b e cylindri able Furthermore we call the subspace B of C a base space Hence
motion planning problems involving a free ying rob ot among fat obstacles have
cylindri able con guration spaces in which the workspace constitutes a valid base
space As a result of the cylindri ability of C it is p ossible to partition the subspace
B into closed regions R or C into cylinders R D such that
jf f j B E f R D gj O
f f
We refer to such a decomp osition of the con guration space C into cylinders as a
constrained cylindri cation The partition of B that leads to the cylinders will b e
called the base partition corresp onding to the cylindri cation Figure illustrates
the terminology The gure shows a three dimensional cylindri able con guration
space C with a two dimensional base space B Hence the rest space D is one
dimensional In addition the gure reveals a fragment of the base partition in the
subspace B and shows the con guration space cylinder R D corresp onding to one
of the regions R in the partition The cylinder in this sp eci c example is b ounded
!
although in many cases e g D IR D IR the cylinder will b e unbounded The
constraints on the base partition guarantee that the cylinder R D is intersected
by at most a constant number of surfaces like f
Let us now consider the con guration space cylinder R D corresp onding to a
region R in a base partition in B By the de nition of a base partition the cylinder
R D is intersected by O constraint hypersurfaces These hypersurfaces sub divide
the cylinder R D into a constant number of cells due to their constant complexity
If we furthermore assume that the cylinders themselves have constant descriptional
complexity achievable by establishing that R has constant complexity then each
of the O free or forbidden cells in R D has constant complexity as well In
conclusion the constraint hypersurfaces and the cylinder b oundaries divide the free
space into constant complexity and thus simple sub cells
f
!
C
R D
D
R
B
Figure A three dimensional example of a cylindri able con guration space C
with a base space B and a fragment of the base partition in B The con guration
space cylinder R D obtained by lifting the base partition region R into C is
intersected by at most a constant number of constraint hypersurfaces f
!
The preceding arguments suggest a two step approach for computing a cell de
comp osition for a motion planning problem with a cylindri able con guration space
rst nd a base partition in some appropriate base space B of C and then tranform
the partition into a cell decomp osition of the free space FP C by computing a
decomp osition of the free part of every cylinder We shall see that the resulting
decomp osition consists of cells that allow for simple motion planning within their
interiors and moreover that the rules for crossing from one cell into another are
simple The prop osed approach follows a pro jection like approach to cell decomp o
sition that is encountered in several other algorithms see e g Basically
these metho ds recursively decomp ose a lower dimensional subspace of the con g
uration space C and lift the decomp osition regions into C The free part of the
resulting cylinders is subsequently partitioned into a number of simple sub cells
In Section it is shown how the latter part of the two step approach outlined
ab ove transforms a base partition into a cell decomp osition of comparable size in
time prop ortional to the size of the base partition Noting this the problem of nd
ing a small cell decomp osition of the free space FP C reduces to the problem of
nding a small sized base partition in an appropriate base space B C Section
exploits sp eci c prop erties of the constraint hypersurfaces that follow from the
CHAPTER A PARADIGM FOR FAT MOTION PLANNING
shap es and relative p ositions of the obstacles to simplify the constraints on the par
tition of the base space B W for motion planning problems involving free ying
rob ots The new and simpler constraints combined with the transformation steps
result in a tailored paradigm for motion planning for free ying rob ots amidst fat
obstacles In the next chapter this paradigm is shown to lead to e cient algorithms
for planning motions for free ying rob ots amidst several types of obstacles in dif
ferent workspaces Moreover the ideas presented in this chapter prove useful for
motion planning problems that do not t neatly in the sketched framework they
lead to an e cient algorithm for planning the motion of a vacuum cleaning rob ot
which is de nitely not free ying
Transforming a base partition into a cell de
comp osition
We consider a motion planning problem for a constant complexity rob ot B amidst
constant complexity obstacles E E Pairs of a feature B and a feature
f f
E of matching dimension de ne constraint hypersurfaces f in the cylindri able
con guration space C B D Furthermore we assume that we are given a graph
V E where V is a set of constant complexity closed regions R that partition
B B B
B and individually satisfy
jf f j B E f R D gj O
f f
and E contains the adjacencies of V s regions E f R R V V j R
B B B B B
R g
The algorithm outlined b elow transforms the graph V E into a connectivity
B B
graph CG V E consisting of a set V of constant complexity sub cells that
C C C
collectively partition the set of free placements FP and a set E f A A
C
V V j A A g of sub cell adjacencies The sizes of the sets V and E
C C C C
are of the same order of magnitude as the sizes of V and E resp ectively jV j
B B C
O jV j jE j O jE j Note that the graph V E supp orts simple path nding
B C B C C
b etween two placements in sub cells A V and A V the constant complexity
C C
of the individual sub cells guarantee easy path nding within a sub cell and the
constant complexity of the shared b oundary of two adjacent sub cells following
from the constant complexity of the involved sub cells caters for simple b oundary
crossing rules The transformation steps are contrary to the computation of the base
partition indep endent of the actual motion planning problem under consideration
Transform Base Partition into Cell Decomposition
V
C
E
C
for all R V do B
TRANSFORMING BASE PARTITION INTO CELL DECOMPOSITION
compute the arrangement A of surfaces f intersecting R D
use A to compute FP R D
Desc R
for all maximal connected comp onents A of FP R D do
V V f Ag
C C
Desc R Desc R f Ag
for all R R E do
! " B
for all A Desc R A Desc R do
! ! " "
if A A then E E f A A g
! " C C ! "
Figure gives a pictorial explanation of the transformation
R
R
R
V E
B B
R
A
CG V E
C C
A
#
A
!
Desc R
A
"
Desc R
Desc R
Desc R
Figure The relation b etween the base partition graph V E in the subspace
B B
B of C at the top and the connectivity graph CG V E in the con guration
C C
space at the b ottom Each no de region R V de nes at most O no des A V
B C
collected in a set D esc R Two no des A and A in V can only b e connected if
C
the corresp onding no des R and R in V are connected so for example A may b e
B !
connected to all no des in D esc R but A and A can never b e connected
! #
We review the di erent steps of the transformation in more detail to verify their
validity and to determine the e ciency Recall that the de nition of the set V
B
and the constant complexity of the regions R V together imply the constant
B
complexity of all sub cells A V C
CHAPTER A PARADIGM FOR FAT MOTION PLANNING
The rst for lo op computes the O constant complexity maximal connected
comp onents of FP R D A p ossible way to compute this free part of the cylinder
R D in constant time could b e to apply the techniques by Schwartz and Sharir
to the constant number of constraint hypersurfaces intersecting the cylinder
R D Each of the four steps in the lo op is easily veri ed to run in constant time
provided that the constraint hypersurfaces f intersecting R D can b e determined
in constant time In future applications of the transformation algorithm we shall
take care that this precondition is ful lled If the requirement is indeed settled
the entire lo op runs in time O jV j Up on termination of the rst lo op each set
B
Desc R stores all no des in V that corresp ond to free sub cells in R D Note that
C
each set Desc R has constant cardinality
Two free sub cells A and A are adjacent if they share a common b oundary
! "
which allows for collision free crossing from one sub cell into the other Such sub
cells A and A can only b e adjacent if their containing cylinders R D A
! " ! !
and R D A are adjacent in C and hence R and R are adjacent in B
" " ! "
An adjacency R R gives rise to only a constant number of adjacencies of no des
! "
A and A in Desc R and Desc R resp ectively due to the constant cardinality
! " ! "
of Desc R and Desc R Two free sub cells A and A in adjacent cylinders are
! " ! "
adjacent if they share a common b oundary Such a common b oundary has constant
complexity since b oth involved free sub cells have constant complexity The nested
for lo op in the second for lo op takes constant time by the ab ove considerations
implying a running time of O jE j for the latter lo op
B
If we combine the time b ounds of the three steps in the paradigm then we nd
that the running time of the entire paradigm dep ends solely on the size of the base
partition in a lower dimensional subspace and on the time to compute the partition
A small and e ciently computable partition is therefore crucial to the success of the
paradigm We notice that the base partition is implicitly sub ject to constraints in
con guration space not more than a constant number of constraint hypersurfaces
may intersect the con guration space cylinder corresp onding to the base partition
region We have however already suggested the p ossibility of using hypersurface
prop erties to translate the constraints into simpler lower dimensional constraints
In the next section we fo cus on the large class of motion planning planning problems
for free ying rob ots amidst k fat obstacles We will see that these problems allow
for a unique choice of base space E cient partitions are likely to b e achievable
in this subspace due to the p ossiblity to translate the implicit con guration space
constraints into simpler constraints in the lower dimensional subspace In Chapter
we shall see that the resulting tailored paradigm really leads to e cient algorithms
for planning motions for free ying rob ots
Finally we mention that the problem of solving a motion planning query nd a
free path from a placement Z Z Z to another placement Z Z Z
! !B !D " "B "D
basically reduces to a p oint lo cation query with Z and Z in V to nd R Z
!B "B B ! !B
and R Z So we need a structure for p oint lo cation in the base space rather
" "B
than in the full con guration space C After having found R and R it takes
! "
A TAILORED PARADIGM FOR FREE FLYING ROBOTS
O time to nd A Z using D esc R and A Z using D esc R followed
by a search in the graph V E for a sequence of sub cells connecting A to A
C C
The constant complexities of the sub cells and of the common b oundaries of pairs of
adjacent sub cells facilitate the transformation of the sub cell sequence into an actual
free path for B
A tailored paradigm for free ying rob ots
We now fo cus on a sp ecial instance of the class of motion planning problems with
cylindri able con guration spaces namely the problem of planning the motion of a
not to o large constant complexity rob ot B with f degrees of freedom moving amidst
n k fat constant complexity obstacles E E where f and k are constants The
restriction on the size of the rob ot is expressed by a b ound on its reach b
B
where b is some p ositive constant and is a lower b ound on the minimal enclosing
hypersphere radii of the obstacles in E For the moment we assume that the rob ot
B do es not self collide that is no part of B can collide with any other part of B
Let O B b e the reference p oint of the rob ot The tailored paradigm presented
b elow suits rob ots with con guration spaces that can b e written as the Cartesian
pro duct of the d dimensional Euclidean workspace W and some other rest space
D of dimension f d
C W D
such that the p osition of the rob ot s reference p oint in the rob ot s workspace is
part of the sp eci cation of its placement A placement Z of the rob ot can thus b e
d
written as Z Z Z where Z W IR and Z D Free ying rob ots
W D W D
t very naturally in this framework Examples for the rest space D are D
for a free ying unsymmetric rigid rob ot in the plane and D for
a similar rob ot in three dimensional space
If either the obstacles are non fat or the rob ot is arbitrarily large the rob ot
B with its reference p oint O xed at some p oint p W may b e able to touch
all obstacles E E The circumstances summarized ab ove however make this
imp ossible the rob ot with its reference p oint xed at p can only touch obstacles
within a distance from the p oint p such obstacles clearly intersect the hypersphere
B
S Theorem implies that the number of obstacles with minimal enclosing
p
B
hypersphere radii at least intersecting any region with diameter b is
B
b ounded by a constant As all obstacles in E have minimal enclosing hypersphere
radius at least the rob ot B can touch no more than O obstacles while its
reference p oint remains xed at p This fact leads to the following lemma which
provides the theoretical feasibility of choosing W as a basis of the cylindrical cell
decomp osition
Lemma For al l p W
jf f j B E f p D gj O
f f
CHAPTER A PARADIGM FOR FAT MOTION PLANNING
Pro of The subspace p D of the con guration space is intersected by constraint
hypersurfaces f A p oint in f p D corresp onds to a placement of the
rob ot B in which its reference p oint is p ositioned at p and its feature touches
an obstacle feature This feature must necessarily b elong to one of the O
constant complexity obstacles that can b e touched by B while its reference p oint
is xed at p Combined with the constant complexity of B itself this implies that
there exist only a constant number of pairs for which f intersects p D
In the sequel we de ne a partition of the workspace that is sub ject to constraints
that are formulated exclusively in the workspace The partition subsequently turns
out to b e a valid base partition for a cylindrical decomp osition of the con guration
space
We de ne the notion of grown obstacles to formalize the observation that the
rob ot B is unable to touch an obstacle E if the distance from the lo cation of B s
reference p oint to the obstacle E exceeds
B
De nition grown obstacle G E
d
Let E be an obstacle in IR and let IR The grown obstacle E is de ned as
d
G E f p IR j d p E g
Note that as an alternative de nition the grown obstacle G E equals the
Minkowski di erence of E and the hypersphere with radius centered at the origin
so
G E E S
O
Clearly the rob ot s reference p oint must lie inside G E in order for the rob ot
B
B to b e in contact with E if the reference p oint lies outside G E there is no
B
danger for B of colliding with E A formalization of these informal observations
leads to a very interesting prop erty on the lo cation of a constraint hypersurface in
con guration space
Lemma Let B and E Then
f f
f G E D
B
Pro of Figure illustrates the construction by means of a two dimensional grown
obstacle G E W IR and a one dimensional rest space D The
B
arguments of the pro of are given in the workspace
Let p p p f such that p W and p D We must prove that
W D W D
p p p G E D which may b e reduced to proving that p G E
W D B W B
since p D is trivially true This means that it should b e proven that the reference
D
p oint of the rob ot B must b e placed inside G E when B s feature touches E s
B feature
A TAILORED PARADIGM FOR FREE FLYING ROBOTS
G E
B
E
B
y
G E
B
x
Figure A grown obstacle and the corresp onding con guration space cylinder for
a rob ot with W IR and C IR
Assume for a contradiction that p G E Then by the de nition of a
W B
grown obstacle the distance from p to E exceeds But then it is imp ossible
W B
for B to reach and touch the obstacle E by the de nition of the reach of a rob ot
In other words no feature B can touch a feature E So the p oint
f f
p p p with p G E cannot lie on f contradicting the assumption
W D W B
of the lemma
The lemma supplies some kind of a simple outer approximation of the lo cation of
a constraint hypersurface in con guration space If a workspace region R do es not
intersect a grown obstacle G E then certainly none of the constraint hypersur
B
faces f with E intersects the con guration space cylinder R D If on the
f
other hand R intersects G E then one or more constraint hypersurfaces f
B
with may but not necessarily must intersect R D As a result the con
f
guration space cylinder R D corresp onding to a region R that is intersected by
O grown obstacles is itself intersected by at most O constraint hypersurfaces
The following de nition of the coverage of a workspace region facilitates a compact
statement of this interesting result
De nition coverage C ov R
d
Let R W IR
C ov R f E E j R G E g
B
Hence C ov R is the set of obstacles E whose corresp onding grown obstacles
G E intersect R The de nition allows for a compact formulation of the preced
B
ing observations regarding the relation b etween the grown obstacles in the workspace
and the constraint hypersurfaces in the con guration space
CHAPTER A PARADIGM FOR FAT MOTION PLANNING
d
Lemma Let R W IR be such that jC ov R j O Then
jf f j B E f R D gj O
f f
Pro of Take a constraint hypersurface f with f R D Now let
E b e such that E By Lemma f G E D Hence necessarily
f B
R D G E D and thus R G E By the de nition of
B B
C ov R and the assumption jC ov R j O it follows that there are only O
obstacles E such that R G E Due to the constant complexity of these
B
obstacles and the rob ot there is only a constant number of hypersurfaces f with
f R D
The lemma states that any region R with jC ov R j O is guaranteed to satisfy
the constraint on the regions of the base partition requiring that the corresp ond
ing cylinder is intersected by O constraint hypersurfaces As a consequence a
decomp osition of the workspace W into regions R with b oth jC ov R j O and
constant complexity is a valid base partition of the base space B W We shall
onstant size coverage refer to workspace partitions of this kind as cc partitions c
c onstant complexity
De nition cc partition
A cc partition V of a workspace W with obstacles E is a partition of W into regions
R satisfying the fol lowing additional constraints
jC ov R j O
R has constant complexity
The constant size coverage constraint jC ov R j O replaces the constraint
jf f j B E f R D gj O the new constraint is
f f
simpler b ecause it is truly a constraint in the workspace The result in Lemma
and the de nition of cc partitions however would b e completely useless if a
partition of W into regions R with jC ov R j O do es not exist Note that the
existence of such a partition solely dep ends on the absence of p oints p W that are
contained in grown obstacles Fortunately such p oints do indeed not exist by
Lemma The result follows immediately from Theorem noting that each
grown obstacle G E is a constant complexity wrapping and by b
B B B
also a constant complexity b wrapping of the obstacle E itself
d
Lemma Let E be a set of n non intersecting k fat obstacles in IR with minimal
enclosing hypersphere radii at least Furthermore let b for some positive
B
constant b Then
a the complexity of the arrangement A G of al l grown obstacle boundaries
G E E E is O n B
A TAILORED PARADIGM FOR FREE FLYING ROBOTS
d
b every point p W IR lies in at most O grown obstacles G E E
B
E
Lemma b shows that it is p ossible to partition the workspace W with the
k fat obstacles of E into constant complexity regions with constant size coverage
d
Notice that the arrangement A G even partitions W IR into O n regions R with
jC ov R j O as each d cell of the arrangement is a subset of the intersection
of O grown obstacles by Lemma b Unfortunately the partition do es not
suit our purp oses b ecause the d cells themselves may have more than constant
complexity Hence it is not a cc partition although it can b e further re ned into
one
In summary we have found Lemma that a cc partition of a workspace with
non intersecting k fat obstacles always exists The cc partition in the workspace
corresp onds by Lemma to a decomp osition of the con guration space into
constant complexity cylinders that are intersected by no more than a constant num
b er of constraint hypersurfaces As a result the cc partition is a valid partition
of the base space W allowing for application of the transformation algorithm from
Section
The algorithm FatMot given b elow combines the search for a small cc partition
with the transformation of that cc partition into a cell decomp osition of the free
space based on the transformation steps from the previous section Besides the cc
partition regions gathered in a set V the rst step is to rep ort the adjacencies of
W
the cc partition regions in a set E and the function C ov V P E mapping
W W
each region R V onto the constant cardinality set of obstacles E E with
W
G E R Occasionally the pair V E will b e referred to as a cc
B W W
partition graph We denote the time required to compute the pair V E as
W W
well as the coverage function C ov by T n where the argument n represents the
number of obstacles in E
The remainder of the algorithm FatMot is a copy of the transformation algorithm
from Section with the exception of the re nement in step The re nement
shows how the precomputed sets C ov R aid in computing in constant time the
arrangement A of all constraint hypersurfaces that intersect the cylinder R D
A closer lo ok at the re nement learns that A is the arrangement of all constraint
hypersurfaces in a set F ff j B C ov R g which is a sup erset of
f f
the set of hypersurfaces f that satisfy f R D Fortunately the easily
computable set F contains only a constant number of hypersurfaces due to the
constant cardinality of C ov R and the constant complexity of B and the individual
obstacles E Crucial to the validity of the approach of computing a somewhat larger
arrangement is the simple observation that A R D i e the restriction of the
arrangement A to the cylinder R D is equivalent to the restriction to R D of
the arrangement of hypersurfaces f with f R D The techniques by
Schwartz and Sharir from may b e useful to compute a decomp osition of the free
part FP R D of a cylinder R D
CHAPTER A PARADIGM FOR FAT MOTION PLANNING
Algorithm FatMot
Find a cc partition graph V E and compute C ov
W W
V
C
E
C
for all R V do
W
F
for all B C ov R do
f f
compute f
F F f f g
compute the arrangement A of all f F
use A to compute a decomp osition of
FP R D into connected sub cells
Desc R
for all constant complexity sub cells A of FP R D do
V V f Ag
C C
Desc R Desc R f Ag
for all R R E do
W
for all A Desc R A Desc R do
if A A then E E f A A g
C C
The re nement of step of the rst for lo op veri es the running time of O jV j
W
for the rst for lo op of the transformation The running time of the entire algorithm
FatMot b ecomes O jV j jE j T n b ecause of the running time of T n for
W W
nding the cc partition graph V E and computing C ov and the O jE j time
W W W
b ound on the execution of the second for lo op see Section The O jV j
W
jE j T n time b ound emphasizes once again that the e ciency of FatMot is
W
fully determined by the size of the graph V E and the time to compute the
W W
graph and the function C ov V P E Since the time T n to compute the
W
graph and the function dominates the time O jV j jE j to simply rep ort b oth
W W
we may conclude that the T n factor dominates the running time of the algorithm
FatMot which may therefore b e said to equal T n
Theorem Let b and k be positive constants In addition let E be a set of k fat
constant complexity obstacles E in the robot s workspace W with minimal enclosing
hypersphere radii at least and let B be a constant complexity robot with reach
b Furthermore let C W D be the con guration space of B Then
B
algorithm FatMot computes a decomposition of the free space FP C into simple
subcells with a connectivity graph CG V E of size O jV j jE j in time
C C W W
O T n where T n is the time to compute a cc partition of the workspace and the
corresponding function C ov V P E
W
Although the exact p erformance of the algorithm dep ends on the ability to nd
small and e ciently computable cc partitions one may at this stage exp ect the
A TAILORED PARADIGM FOR FREE FLYING ROBOTS
metho d to b e rather e cient since the paradigm reduces the problem of nding a
decomp osition of certain f cells in an arrangement in f dimensional con guration
space to the problem of nding some constrained partition of the d dimensional
workspace d f Besides the dimensional reduction there is also the feeling
that the hypersurfaces in con guration space have more complex shap es than the
obstacles in the workspace that are resp onsible for the partition constraints A
ma jor part of the next chapter is devoted to providing convincing and less intuitive
arguments for the validity of our approach simply by deducing small and e ciently
computable cc partitions for workspaces with various kinds of obstacles Another
part of that chapter exploits the more general ideas of this chapter on cylindri able
con guration spaces to obtain an e cient algorithm for planning the motion of a
non free ying rob ot amidst fat obstacles
In Chapter we have seen that within our framework self collisions have no
ma jor implications for the complexity of the free space The O self collision con
straint hypersurfaces do not increase the asymptotic complexity of the arrangement
of constraint hypersurfaces and hence of the free space Let us now reveal the
consequences of self collisions for the motion planning paradigm given as the al
gorithm FatMot We have seen that self collisions are indep endent of the p osition
of the rob ot s reference p oint as they can o ccur anywhere in the workspace The
corresp onding constraint hypersurfaces in C W D are therefore of the form
W s where s D As a result any self collision hypersurface f intersects al l
s
con guration space cylinders R D The constant number of such hypersurfaces
and their individual constant complexity however guarantee that the combinatorial
complexity of the arrangement A and the set FP R D in steps and of the
algorithm FatMot and the running time of FatMot are not a ected The correctness
of the paradigm after the incorp oration of self collisions is established by replacing
the initialization of F in step by the initialization F F where F is the
s s
set of all self collision constraint hypersurfaces
! CHAPTER A PARADIGM FOR FAT MOTION PLANNING
Chapter
E ciently computable base
partitions
This ob jective in this chapter is to nd instances of the general paradigm presented in
Chapter for a handful of di erent settings of the motion planning problem Besides
a universal and nearly optimal solution for planning in two dimensional workspaces
we shall consider four di erent problems in three dimensional workspaces
Appropriate workspace decomp ositions for application of the algorithm FatMot
are shown to exist for problems involving a free ying rob ot moving among p oly
!
hedral and arbitrary obstacles The decomp ositions have sizes O n and O n
!
and are computable in O n log n and n time resp ectively FatMot transforms
the workspace partitions into cell decomp ositions of asymptotically equivalent size
Sections and discuss the details of the resp ective partitions and their con
struction
Nearly optimal results are obtained for two classes of motion planning problems
in space with regularly encountered additional prop erties The rst class consists
of problems involving a free ying rob ot and arbitrary obstacles from a b ounded
range of sizes More precisely the ratio of the minimal enclosing hypersphere radii
of any pair of obstacles is b ounded by a constant The workspace with the obstacles
of this type of problem allow for a simple cc partition of size O n Section re
p orts the details of the partition and its computation The second class of problems
discussed in Section concerns a further constrained rob ot The rob ot s reference
p oint is con ned to a plane in the workspace W The class contains the realistic
problems where the rob ot moves on a work oor Such problems are often encoun
tered in industrial environments One example is the vacuum cleaner rob ot studied
in Contrary to all other problem types dealt with so far the workspace W is
not a pro jective subspace of the rob ot s con guration space C Clearly the rob ot
is not free ying Thus the paradigm of Section do es not apply directly to
this class of problems The plane to which the rob ot s reference p oint is con ned
however is a subspace of C and it turns out that this plane is decomp osable into
regions such that the free part of the con guration space cylinders obtained after
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
lifting the regions into C has constant complexity The algorithm in Section
transforms the decomp osition into an actual cell decomp osition of the free space
The base decomp osition in the plane very strongly resembles the decomp osition
outlined b elow in Section
Arbitrary obstacles in space
In order to get a feeling for the di erent asp ects of computing an appropriate base
partition we rst fo cus on planar motion planning b efore we move on to three
dimensional workspaces and obstacles The aim in this section is to nd a small cell
decomp osition of the free space for the class of motion planning problems with the
following characteristics
A constant complexity rob ot B with f degrees of freedom f with
reach moves freely in the workspace W IR amidst a collection E
B
of k fat constant complexity obstacles E W with minimal enclosing
circle radii at least for some constant k The system is constrained
by the inequality b for some xed constant b
B
A substantial part of the sp eci cation of a placement of the free ying rob ot B is
the p osition of its reference p oint O B in the workspace B As a result the
con guration space C of the problem is the Cartesian pro duct of the dimensional
Euclidean workspace W and some f dimensional space D hence C W D
IR D For a rigid rob ot f the space D equals the one dimensional rotational
interval for free ying articulated rob ots f the space D mo dels the
relative placements of the rob ot s links
The partition that is prop osed b elow a vertical decomp osition of the arrange
ment of grown obstacle b oundaries is a simple example of a conceptually two layer
approach that we will use more often in this chapter This two layer approach nds
a partition of W by rst dividing W into regions with constant size coverage and
subsequently re nes the regions to obtain constant complexity regions Notice that
the instance of algorithm FatMot presented in this subsection provides a general
ization of the algorithm presented in Section which is a mo di ed version of
Schwartz and Sharir s algorithm and as such dedicated to a p olygonal rob ot
amidst p olygonal obstacles
Our rst step towards a cc partition comprises the computation of the grown
obstacle b oundaries G E for all obstacles E E Each b oundary is obtained
B
in O time leading to O n time for computing all b oundaries As a preparation
for the next step each grown obstacle b oundary G E is cut up into O
B
arcs which are maximal connected x monotone b oundary parts having no vertices
in their interiors Note that the arc endp oints are generally incident to two arcs
For future purp oses we lab el each arc from G E with E Lemma rep eats
B
earlier results on the arrangement A G in a formulation that b etter suits their
ARBITRARY OBSTACLES IN SPACE
present application The b part follows directly from Lemma if one realizes
that all p oints p in a single cell A A G lie in exactly the same collection of
grown obstacles
Lemma Let A G be the planar arrangement of al l boundaries G E E
B
E Then
a A G has complexity O n
b jC ov A j O for al l faces A A G
By Lemma the resulting O n arcs de ne only O n yet unknown arc inter
sections and additionally sub divide W into regions with constant size coverage
In a second step we compute the vertical decomp osition of the arrangement
A G of grown obstacle b oundaries by sweeping the plane with the arcs with a
vertical line meanwhile extending walls in upward and downward vertical direction
from all O n arc endp oints known in advance and all O n arc intersections to
b e determined during the sweep Figure shows an example of the vertical
decomp osition of an arrangement of x monotone arcs For simplicity we assume
Figure The vertical decomp osition of an arrangement of x monotone arcs walls
are extended in vertical direction from all arc endp oints and arc intersections
that no two events p oints that is arc intersections or arc endp oints lie on a vertical
line x X The extended walls end on the rst arc that is hit in the direction of the
extension The walls sub divide the cells of the arrangement into regions b ounded
by two p ossibly degenerate vertical walls and two arc sections Hence the regions
in the vertical decomp osition of the arrangement A G have constant complexity
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
Moreover they inherit the constant size coverage from the original enclosing cell
of A G The arcs and their endp oints and pairwise intersections and the walls
and their endp oints collectively form a planar graph consisting of O n edges and
vertices sub dividing the plane into O n constant complexity regions the regions
of V The set E of pairs of adjacent vertical decomp osition regions has size O n
W W
as well as each adjacency can b e charged to one of the O n edges in the planar
graph Moreover it is clear that all p ossible sets E and V also have size n
W W
Lemma V is a cc partition of size n of W with the obstacles E E
W W
f R R V V j R R g has size n
W W
The plane sweep algorithm must not only rep ort the regions of V but also the cov
W
erages C ov R of all regions R V and the region adjacencies of E To achieve
W W
this the following invariant regarding the available data is maintained throughout
the entire sweep
V contains all regions strictly left of the sweep line and their descriptions
W
E contains all adjacencies of regions left of the sweep line C ov R is assigned
W
for all regions R currently in V
W
The sweep line status is a top to b ottom ordered cross section of the vertical
decomp osition of A G at x X As such it is an alternating sequence of in
tersected regions and intersected arcs The elements of the sequence are stored
in the leaves of a balanced binary tree The regions in the data structure are
accompanied by their coverages and partial descriptions i e their p ossibly
degenerate left vertical b ounding wall and upp er and lower b ounding arc
The event point schedule is the sequence of up coming events consisting of
all statically computable arc endp oints and p otential intersections of pairs
of consecutive arcs separated by a single region in the sweep line status
The summarized events are stored in a priority queue ordered by increasing
x co ordinate The up coming event is always either an arc endp oint of an
intersection of two consecutive arcs so the rst event in the queue is indeed
the up coming event
The creation of an arti cial event p oint at x and the prop er initialization
of the event p oint schedule and the sweep line status causes the consistent main
tainance of the invariant to eventually lead to the situation where the event p oint
schedule is empty the set V consists of all regions in the vertical decomp osition of
W
A G the set E contains all pairs of adjacent region from V and the function
W W
C ov is assigned appropriately for all arguments R V
W
Figure shows the di erent kinds of events that are encountered a c show
all p ossible endp oint events d shows the intersection event Notice that the end
p oint events are vertices joining two arcs originating from a single grown obstacle
ARBITRARY OBSTACLES IN SPACE
U U
u u
Q R
g
f
Q
R R Q
g
f
Q
R
l
l
L L
a b
U U
u u
R Q
R Q
f g
g R Q
f
f g
R Q
Q
R
l
l
L L
c d
Figure The di erent types of event p oints
b oundary and hence carrying the same lab el The dotted lines are the vertical
decomp osition walls extended from the event p oints
The up coming event can b e extracted in constant time from the event p oint
schedule The event p oints and more particularly the walls extended from it mark
the end of at most three consecutive regions in the sweep line status named R R
R in Figure The ending regions and their descriptions which are completed
by the addition of the right b oundary a wall or event p oint are deleted from
the balanced binary tree storing the sweep line status and transferred to V The
W
corresp onding coverages are assigned to the appropriate entries of C ov The sepa
rating arcs f f are deleted along with the ending regions The deleted regions
are replaced by at most three new regions named Q Q Q in Figure and
their separating arcs The regions are accompanied by their partial representations
which involve the walls the new separating arcs g g and the two arcs u and l
b ounding the upp er new region from ab ove and the lower new region from b elow
The identi cation of the latter two arcs requires two searches of the sweep line sta
tus The coverages of the new regions are easily computable from the coverages
of the old regions using the simple observations that the coverages of regions on
opp osite sides of a wall are equal and the set di erence of the coverages on opp osite
sides of an arc with lab el E is exactly fE g Finally we must rep ort the adjacen
cies of the new regions These adjacencies only involve the O new regions the
O old regions and the regions U b ounding the old and new upp er regions from
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
ab ove and L b ounding the old and new lower regions from b elow The latter two
regions are found by two searches of the binary tree storing the sweep line status In
summary the pro cessing of an event requires a constant number of searches dele
tions and insertions in a balanced binary tree The data structure supp orts each of
these op erations in time O log n The remaining computations in a single step a
constant complexity data set and therefore require constant time
The O newly obtained pairs of consecutive arcs in the sweep line status in
volving the new separating arcs and u and l may necessitate an up date of the event
p oint schedule with the p otential intersections of the new consecutive arc pairs The
insertion of an element in the priority queue storing the schedule takes O log n time
As a result the entire up date of the event p oint schedule takes O log n time
Throughout the plane sweep a total of O n events are encountered each requir
ing O log n pro cessing time The entire sweep and hence the computation of the
sets V and E and the function C ov therefore takes O n log n time
W W
Lemma The computation of the cc partition graph V E and the corre
W W
sponding function C ov V P E takes O n log n time
W
Lemma shows that the time T n in Theorem to compute the cc partition
graph V E and the function C ov is O n log n In addition Lemma b ounds
W W
the sizes of the sets V and E by O n Hence the algorithm FatMot computes
W W
a decomp osition of FP of size O jV j jE j O n in time T n O n log n
W W
Note that the outlined vertical decomp osition must b e substituted for the rst step
of the algorithm in order to actually achieve this p erformance
Theorem Let k and b be constants and let E be a collection of k
fat constant complexity obstacles E W IR with minimal enclosing circle radii
at least Algorithm FatMot solves the motion planning problem for any constant
complexity robot B with f degrees of freedom and reach b amidst E in time
B
O n log n The connectivity graph CG V E of the resulting decomposition of
C C
FP into simple subcells has optimal size O n
Polyhedral obstacles in space
In this section we move on to three dimensional workspaces where we study a
setting of a free ying rob ot amidst p olyhedral obstacles The number of algorithms
for motion planning problems in a three dimensional workspace with p olyhedral
obstacles is limited The two metho ds that apply to rob ots with an arbitrary number
f !
f of degrees of freedom are the general O n cell decomp osition algorithm by
f
Schwartz and Sharir and the O n log n roadmap metho d by Canny More
sp eci c results include O n and O n log n algorithms for a DOF
ladder among p olyhedral obstacles and an O n algorithm for a p olyhedral
rob ot in the same environment This section presents an instance of the algorithm
POLYHEDRAL OBSTACLES IN SPACE
FatMot for a b ounded size rob ot among fat p olyhedral obstacles with running time
O n log n regardless of the number f of degrees of freedom of the rob ot The
following description xes the setting of the results
A constant complexity rob ot B with f degrees of freedom f with
!
reach moves freely in the workspace W IR amidst a collection E
B
of k fat constant complexity p olyhedral obstacles E W with minimal
enclosing sphere radii at least for some constant k The system is
constrained by the inequality b for some xed constant b
B
The problem of nding cc partitions for three dimensional Euclidean workspaces
is much harder than its two dimensional equivalent which is illustrated by the
relatively small number of results on partitioning D arrangements into constant
complexity sub cells like tetrahedra or prisms Moreover the existing results see
for example pap ers by Aronov and Sharir Chazelle and De Berg Guibas
and Halp erin do not apply to arbitrary arrangements but instead only hold
for arrangements of planar faces which makes their application to the arrangement
of arbitrarily shap ed grown obstacles imp ossible The two step approach of rst
decomp osing the workspace into constant size coverage regions and then re ning
the regions to constant complexity regions is likely to lead to O n regions as the
application of any of the ab ove metho ds gives O n sub cells when applied to an
O n complexity arrangement of planar faces Although a smaller decomp osition
might b e achievable by either this approach or a completely di erent strategy we
are currently unaware of such a decomp osition and therefore choose to settle for
a cc partition of size O n The partition is obtained by following the two step
approach
Instead of using the grown obstacle b oundaries to achieve the decomp osition
into constant size coverage regions we now use the b oundary of a p olyhedral outer
approximation of these grown obstacles The p olyhedral approximations still achieve
a decomp osition of W into regions of constant size coverage but additionally allow
for subsequent application of a vertical decomp osition algorithm to the triangulated
p olyhedral arrangement to obtain O n constant complexity regions A tight outer
approximation of the grown obstacle G E is the Minkowski di erence H E
B B
of E and a cub e with side length
B
De nition Let S O be some arbitrary rotation matrix establishing that
none of the faces of the cube C is vertical Then
O
B
H E E C
B O
B
The computation of a Minkowski di erence H E from the constant complexity
B
obstacle E takes constant time The Minkowski di erence H E encloses E
B
p
and by its de nition no p oint in H E has a distance larger than to
B B
p
wrapping of E and b ecause b also a E Hence H E is a
B B B
!" CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
p
wrapping of E The additional lower b ound of on the minimal enclosing b
sphere radii of all obstacles in E makes Theorem applicable to the arrangement
A H of all Minkowski di erence b oundaries H E E E Thus we obtain
B
Lemma
Lemma Let A H be the three dimensional arrangement of al l boundaries
H E E E Then
B
a A H has complexity O n
b jf E E j H E A gj O for al l faces A A H
B
The set of obstacles E whose Minkowski di erences H E intersect a given region
B
R is a sup erset of C ov R the set of obstacles E whose grown obstacles G E
B
intersect R due to the inclusion G E H E With the observation we
B B
deduce the following interesting corollary from Lemma b
Corollary jC ov A j jf E E j G E A gj O for al l faces
B
A A H
Hence the O n complexity p olyhedral arrangement A H sub divides W IR into
regions with constant size coverage
The range searching results in Chapter facilitate a computation of the linear
!
complexity arrangement A H in time O n log n log log n A naive and simpler
but in the light of steps to come su ciently e cient computation of A H takes
!
O n time and simply intersects all pairs of constant complexity faces of Minkowski
di erence b oundaries H E and stores the p otential intersection segment with
B
b oth faces After that each face and the segments de ned by its intersection with
other faces undergo a constrained triangulation The triangulation is constrained
in the sense that it incorp orates all intersection edges as edges of triangles in the
triangulation The triangulation introduces no new vertices The constrained trian
gulation can b e done by a single sweep comparable to the sweep in Section of
each face f in time O m log m where m is the complexity of the resp ective face
f f f
P
and the corresp onding intersection segments As the cumulative complexity m
f
f
equals asymptotically the complexity O n of the arrangement A H the triangu
lation of all faces of the arrangement takes O n log n time The result is a collection
T of non intersecting triangles Although the triangles are non intersecting they
A H
do touch each other that is they share edges and vertices For future purp oses
we take care to lab el each triangle t T with the appropriate obstacle E to
A H
indicate that t b elongs to the Minkowski di erence H E The decomp osition
B
of the workspace by the arrangement A H is not a cc partition b ecause the cells
of the arrangement may have more than constant complexity To re ne the cells
into constant complexity regions we apply a full vertical decomp osition algorithm
to the triangulated arrangement
POLYHEDRAL OBSTACLES IN SPACE
De Berg Guibas and Halp erin give a rather simple algorithm for computing
a full vertical decomp osition of an arrangement of triangles in space The general
p osition of the obstacles in E and the rotated cub e C establish that the
O
B
triangles in T are in general p osition in the sense that no triangle is vertical
A H
no edge is parallel to a co ordinate axis and no two edges lie in a vertical plane
unless they coincide The fact that we deal with sets of touching triangles requires
some additional b o okkeeping to prevent multiple extensions of equivalent walls The
b o okkeeping is simple and do es not a ect the e ciency of the algorithm We neglect
the b o okkeeping in the description of the algorithm In the restricted case of non
intersecting triangles the vertical decomp osition algorithm leads to a decomp osition
of the arrangement of the triangles into O n constant complexity regions
The computation takes O n log n time Below we rst brie y discuss the
algorithm and the structure of the full vertical decomp osition After that we show
that application of the algorithm to the set of triangles T leads to a cc partition
A H
graph V E with jV j O n but unfortunately a larger set E To remedy
W W W W
this we will replace each triangle by a at tetrahedron and then show that the
resulting set F T of triangular tetrahedron faces solves the problem as its full
A H
vertical decomp osition leads to jV j jE j O n and T n O n log n
W W
The computation of the full vertical decomp osition pro ceeds in two steps The
rst step results in the vertical decomp osition of the arrangement of non intersecting
triangles The second step uses the sp eci c shap e of the resulting regions to sub di
vide them further to obtain constant complexity regions which together constitute
the full vertical decomp osition We discuss each step in more detail
The vertical decomp osition step partitions the arrangement of non intersecting
triangles from a given set T into maximal connected collections of p oints with equal
vertical visibility with resp ect to the triangles of T b oth in upward and downward
z direction More precisely a region in the vertical decomp osition is a maximal con
nected set fx IR jup x t down x t g where t t T and up x down x
denotes the rst triangle in T that is hit by the vertical ray emanating from x in
upward downward z direction The decomp osition is achieved by the extension of
vertical walls from all triangle edges which end up on hitting other triangles The
representation of the vertical decomp osition computed by the vertical decomp osition
algorithm consists of i for each triangle edge e the wall W e extended from it
and ii for each triangle t T the two arrangements of ending walls on either side
of t The algorithm stores the walls and the triangle arrangements in a quad edge
structure to facilitate future navigating through the decomp osition and explicit
rep orting of the regions and the region adjacencies
The wall W e extended from the edge e is obtained by intersecting the vertical
surface H e through e that is the union of all vertical lines through e with all
non intersecting triangles resulting in O n disjoint intersection segments in H e
The upp er b oundary of W e is de ned by the lower envelope of all intersection
segments in H e lying ab ove e Similarly the lower b oundary of W e is de ned by
the upp er envelope of all intersection segments in H e lying b elow e The envelopes
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
are alternating p olygonal chains of parts of intersection edges and of vertical seg
ments Both envelopes have complexity O n since the upp er lower envelope of a
collection of disjoint line segments in the plane has complexity O n Each halfwall
b etween an envelope and the edge is divided into slabs by vertical line segments
connecting the vertices of the envelope to the edge see Figure The vertical seg
e
Figure The grey face is the upp er half of the wall extended in vertical direction
from the edge e The line segments ab ove e are the intersections of triangles with
the supp orting plane of the vertical surface H e
ments corresp ond to intersections with other walls Clearly the vertical segments
do not increase the complexity of the wall The envelopes and the vertical line
segments can b e computed in O n log n time using divide and conquer The
computation of all O n walls requires O n log n time the cumulative complexity
of the walls is O n
Walls end as line segments on b oth sides of the triangles of T The ending walls
on the upward facing side of the triangle are non vertical p ortions of the lower
b oundaries of walls W e Similarly the ending walls on the downward facing side
are p ortions of upp er b oundaries of walls W e Figure shows an example of a
triangle side and the arrangement of walls ending on it By charging the complexity
of the arrangement of ending walls to the corresp onding walls and by noticing that
the complexity of T is O n we nd that the asymptotic cumulative complexity of
all triangle arrangements equals the cumulative complexity of all walls O n A
single scan of all walls su ces to nd for all triangle sides the segments that de ne
the arrangement on that sp eci c side Next a single arrangement can b e computed
by a single sweep of the segments in time O m log m where m is the complexity of
POLYHEDRAL OBSTACLES IN SPACE
t t
Figure The arrangement of walls ending on a side of a triangle and the planar
vertical decomp osition of the arrangement
the arrangement As the cumulative complexity of all arrangements is O n the
computation of all arrangements takes O n log n time The o ors and ceilings
that is the b ounding faces in vertical direction of the resulting regions are parts of
triangles More precisely they are faces in an arrangement of ending walls on a
triangle side Note that the o or and ceiling of a vertical decomp osition region have
equivalent pro jections onto the x y plane
The second step re nes the vertical decomp osition into a full vertical decomp o
sition consisting of regions b ounded by six p ossibly degenerate planar faces by
adding walls parallel to the x z plane The re ning is obtained through a planar
vertical decomp osition of all triangle arrangements in which segments are extended
within the triangles parallel to the y axis from every vertex of the arrangement see
Figure The additional walls connect corresp onding extended segments that is
with equivalent pro jections on the x y plane in the upp er and lower b ounding tri
angles of a vertical decomp osition region The entire re ning takes O n log n using
a sweep of all triangle arrangements Every region in the full vertical decomp osition
has a trap ezoidal o or and ceiling with equivalent pro jections onto the x y plane
The remaining four p ossibly degenerate b ounding faces are vertical walls two re
sulting from the vertical decomp osition and two added during the re nement The
representation of the full vertical decomp osition computed in the re nement step
consists of i all walls W e extended from triangle edges e plus all additional
paralleloid walls parallel to the x y plane and ii for each triangle t T the
two arrangements of ending walls b oth extended from edges and added during the
re nement on either side of t
!
Lemma Let T be a set of n non intersecting triangles in IR The ful l vertical
decomposition of T decomposes the arrangement of triangles in T into constant
complexity regions The decomposition has complexity O n and can be computed
in time O n log n
Application of the decomp osition algorithm to the set of triangles T yields
A H
a sub division of the constant size coverage faces of the arrangement A H into
constant complexity regions As a result the regions of the re ned sub division
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
collectively partition W with the p olygonal obstacles E into O n regions with
constant complexity and constant size coverage Hence the collection of these re
gions would make an appropriate choice for a set V Unfortunately the decomp o
W
sition do es not give us a low number of adjacencies as well
The common b oundary of two adjacent regions in the full vertical decomp osition
is either embedded in a vertical wall or embedded in a triangle Although the
number of adjacencies of the rst type can b e b ounded by O n it seems imp ossible
to obtain a similar b ound on the number of adjacencies of the latter type Walls
extended from other triangles edges end on b oth sides of a triangle t T and
A H
de ne arrangements of line segments on these sides The complexity of a single
arrangement and its planar vertical decomp osition can b e as high as O n and
hence the number of faces in the vertically decomp osed arrangement is b ounded by
O n only These faces are the o ors or ceilings of the full vertical decomp osition
regions Each non empty intersection of faces on either side of a single triangle
corresp onds to an adjacency of two regions In general two sub divisions of a single
triangle t into m and n faces could easily give rise to O m n non empty
t t t t
intersections of pairs of faces At present it is unclear if the sp eci c prop erties of
the full vertical decomp osition make it p ossible to b ound the cumulative number of
intersections on all O n triangles and hence the number of region adjacencies by
anything close to O n
An elegant way to overcome the problem outlined ab ove is by replacing all tri
angles of T by tetrahedra that are su ciently at to prevent them from inter
A H
secting The tetrahedra have the initial triangles of T as one of their faces
A H
The triangular faces of the tetrahedra are collected in the set F T F T
A H A H
satis es F T T and jF T j j T j Hence the size of F T
A H A H A H A H A H
is still O n Moreover F T is again a set of non intersecting though touching
A H
triangles which now have the simple but b ene cial prop erty that one of their sides
faces the interior of a tetrahedron consisting of four triangles from F T
A H
The b ene t of this prop erty lies in the fact that walls extended from triangles out
side are unable to p enetrate and hence to end on the inward facing sides of its
triangular faces as they end up on hitting the outside of The walls inside must
therefore either b e extended from one of the six edges of itself or added during
the subsequent re nement of the O regions in the vertical decomp osition inside
Clearly the re nement introduces no more than a constant number of walls as
well As a result the O ending walls on the inward facing side of a triangle de ne
a constant complexity arrangement on that side
We rst discuss how to replace each triangle t T by a tetrahedron having
A H
t as one of its faces such that the resulting tetrahedra are non intersecting Recall
that T is a set of triangles that may share a vertex or an edge but do not
A H
intersect each others interiors Let b e the minimum distance b etween any pair
of disjoint non touching triangles Furthermore let b e the minimum over all
dihedral angles b etween pairs of touching triangles that share an edge and b e
the minimum over all angles b etween the supp orting plane of a triangle t T
A H
POLYHEDRAL OBSTACLES IN SPACE
and an edge of another triangle t incident to a vertex of t We de ne min
We construct a set F T by applying the following pro cedure to every t T
A H A H
Let v v v b e the vertices of t
The planes through v v v that make a p ositive angle with the
top side facing z of t intersect in a p oint v If the distance from v to t is
strictly less than then the tetrahedron is de ned by the vertices v v v v
If the distance from v to t is at least then we take the half line h through v
and p erp endicular to and ending on t The tetrahedron is de ned by v v v
and the unique p oint v on the half line h with distance to t
The application of the ab ove two step pro cess to the triangles of T results in a
A H
collection of tetrahedra with disjoint interiors Prop erty is a compact statement
of the result The prop erty contains a minor abuse of the de nition of the set
F T as it is interpreted to b e the set of at tetrahedra instead of the triangular
A H
faces of the tetrahedra Let the op en interior of the closed set b e denoted by
int
Prop erty F T int int
A H
Before applying the vertical decomp osition algorithm to the triangles of F T
A H
we must convince ourselves that these triangles partition the workspace W with the
obstacles E into regions with constant size coverage Fortunately this follows di
rectly from the construction of F T from the triangles of T which already
A H A H
de ne a partition of into regions with constant size coverage The addition of dis
joint triangles to the partition can only lead to a re nement of the regions into
smaller regions with smaller or equally sized coverages
Application of the decomp osition algorithm to the O n disjoint triangles of
F T yields a full vertical decomp osition of complexity O n and therefore
A H
consisting of O n regions with constant complexity The decomp osition regions
are app ointed to b e the regions of V The coverage of each region R V has
W W
constant size as R is a subset of a region in the partition by the triangles of F T
A H
which were shown to have constant size coverage in the previous paragraph
Let us now b ound the size of the set E f R R V V j R R g
W W W
The complexities of the triangle arrangements are crucial to the analysis of the
adjacencies so we rst study these complexities in more detail The complexity of
the entire full vertical decomp osition is O n As a consequence the cumulative
complexity of all triangle arrangements is O n as well Each triangle t F T
A H
has a side facing the interior of the tetrahedron it b elongs to and a side facing
outward The complexity m of the arrangement on the inward facing side of t is
t
constant b ecause we have seen that only a constant number of walls de ne this
arrangement m O for all t The complexity n of the arrangement on the
t t
outward facing side of a single triangle t however can b e as high as O n The
number of adjacencies of faces in a triangle arrangement is of the same order of
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
magnitude as the complexity of the arrangement b ecause a triangle arrangement
is a planar graph Hence the numbers of adjacencies on the inward and outward
facing sides of a triangle t are O m and O n resp ectively
t t
We recall that two adjacent regions share a common b oundary that is either
embedded in a triangle or embedded in a vertical wall Each non empty intersection
of two faces in arrangements on either side of a triangle represents an adjacency of
the rst type The number of non empty intersections on opp osite sides of a triangle
t is O m n O n due to m O Hence the number of adjacencies in E
t t t t W
P
of the rst type equals O n O n To nd the number of adjacencies of
t
t
the second type note that two adjacent regions whose common b oundary is part of
a vertical wall have adjacent o ors or adjacent ceilings in a triangle arrangement
Hence the total number of adjacencies of faces in all triangle arrangements supplies
an upp er b ound on the number of pairs of regions in V that share a vertical
W
face The total number of adjacencies of faces on a triangle t is O m n
t t
O n Hence the number of adjacencies in E of the second and last type equals
t W
P
O n O n as well Lemma summarizes the b ounds on the sizes of V
t W
t
and E
W
with the polyhedral Lemma V is a cc partition of size O n of W IR
W
obstacles E E f R R V V j R R g has size O n
W W W
After applying the O n log n vertical decomp osition algorithm we traverse the
O n constant complexity regions of the decomp osition using the quad edge struc
ture storing the triangle arrangements and the walls starting from an arbitrarily
chosen region The aim of the traversal is to extract explicit descriptions of the
regions in V rep ort the region adjacencies of E and compute the coverages
W W
C ov R of the regions R V The latter part of the computation deserves some
W
additional explanation Instead of attempting to compute the constant size cover
ages C ov R fE E j G E R g directly we rst compute the constant
B
cardinality sets fE E j H E R g The constant cardinality of these set is
B
due to the prop erty that each region R V is a subset of a face A of the arrange
W
ment A H which satisfy jf E E j H E A gj O by Lemma The
B
necessary data for the computation of the sets fE E j H E R g are avail
B
able from the decomp osition contrary to the data for the computation of C ov R
Throughout the traversal of the decomp osition we use the fact that adjacent re
gions are intersected or actually enclosed by the same set of Minkowski di erences
H E unless their common b oundary is contained in a triangle t that is part of
B
some Minkowski di erence b oundary H E in which case the sets of intersect
B
ing Minkowski di erences di er by exactly fE g the lab el of the triangle t The set
fE E j H E R g is a sup erset of C ov R fE E j G E R g
B B
since G E H E The latter set is obtained in constant time from the
B B
former set by elimination of the O obstacles E f E E j H E R g that
B
satisfy E G E The traversal of the O n regions in the decomp osition
B requires taking into account the limited amount of work p er traversed region time
ARBITRARY OBSTACLES IN SPACE
prop ortional to the number of regions As a consequence the time to compute the
cc partition graph V E and the function C ov is dominated by the running time
W W
of the vertical decomp osition algorithm O n log n
Lemma The computation of the cc partition graph V E and the corre
W W
sponding function C ov V P E takes T n O n log n time
W
Substitution of the computation of V E and C ov outlined ab ove for the
W W
rst abstract step of the algorithm FatMot yields by Theorem and Lemmas
and a motion planning algorithm with running time O n log n The
algorithm decomp oses the free space of a rob ot amidst fat p olyhedral obstacles into
O n sub cells of constant complexity de ning O n pairwise adjacencies
Theorem Let k and b be constants and let E be a collection of k fat
constant complexity obstacles E W IR with minimal enclosing sphere radii
at least Algorithm FatMot solves the motion planning problem for any constant
complexity robot B with f degrees of freedom and reach b amidst E in time
B
O n log n The connectivity graph CG V E of the resulting decomposition
C C
of FP into simple subcells has size O n
The gap b etween the linear complexity of the free space and the quadratic size
of the connectivity graph of the FP decomp osition shows that the cell decomp osition
is not optimal An interesting op en problem is therefore to attempt to bridge the
gap by exploring alternative cc partitions of the workspace A smaller cc partition
would probably require a partitioning strategy that di ers completely from the two
step approach of rst sub dividing the workspace with the obstacles into regions with
constant size coverage and then re ning the regions in the sub division to constant
complexity regions The next two sections show examples of cc partitions that are
not obtained via the two step approach
Arbitrary obstacles in space
The setting that is considered in this section di ers from the setting of the previous
section in that the obstacles are not required to b e p olyhedral but instead only
assumed to b e of constant complexity The only general metho ds that could solve
such a problem are those by Schwartz and Sharir and Canny Here it is
shown that an instance of the algorithm FatMot for a b ounded size rob ot among
fat obstacles exists with running time n indep endent of the actual number f of
degrees of freedom of the rob ot The setting is xed by the following description
A constant complexity rob ot B with f degrees of freedom f with
reach moves freely in the workspace W IR amidst a collection
B
E of k fat constant complexity obstacles E W with minimal enclos
ing sphere radii at least for some constant k The system is
constrained by the inequality b for some xed constant b B
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
The main implication of the fact that the obstacles in the workspace have arbi
trary shap e is that they cannot b e tightly wrapp ed by some p olyhedron of constant
complexity A consequence is that we are no longer able to construct a low complex
ity arrangement in a rst decomp osition step which partitions the workspace into
regions with constant size coverage Instead we prop ose a simple direct cc partition
of the workspace leading to a cubic number of regions
The axis parallel b ounding b oxes B G E of the grown obstacles G E
B B
E E partition the workspace with the obstacles E into regions with constant cov
erage which is not completely obvious The arrangement of b oxes has complexity
O n Unfortunately the cells of the arrangement may very well have more than
constant complexity If however we replace the arrangement of b ounding b oxes
by the arrangement of the supp orting planes of all b ounding b ox faces then we
obtain rectangloid constant complexity cells without increasing the asymptotic
worst case complexity of the arrangement the cc partition of the workspace by the
supp orting planes has complexity n
Each obstacle E E contributes six planes to the arrangement de ning the
cc partition The constant complexity of the obstacle E allows us to compute the
supp orting planes x x x x y y y y z z and z z of the
grown obstacle G E in constant time For simplicity we assume that none of
the supp orting planes is tangent to any other grown obstacle G E E E
B
This extra assumption however can b e avoided quite easily After having computed
all n planes parallel to the y z plane we sort the resulting planes by increasing
x co ordinates yielding a sequence x x The sequence partitions the real
n
line into n intervals X h n with X x X x x
h h h h
for all h n and X x A similar treatment of the planes
n n
parallel to the x z plane and x y plane results in sequences y y and
n
z z and two partitions of the real line into intervals Y i n
n i
and Z j n resp ectively The strictly increasing nature of the sequences is
j
due to the assumption that no plane is tangent to two grown obstacles The three
ordered sequences de ne a cc partition graph consisting of a set V of regions
W
V f X Y Z j h i j n g
W h i j
and a set E of adjacencies
W
E f X Y Z X Y Z j h n i j n g
W h i j h i j
f X Y Z X Y Z j i n h j n g
h i j h i j
f X Y Z X Y Z j j n h i n g
h i j h i j
Lemma V is a cc partition of size n of W with the obstacles E E
W W
f R R V V j R R g has size n
W W
ARBITRARY OBSTACLES IN SPACE
Pro of V and E are easily seen to have size n The remaining task is to
W W
prove that the regions of V partition the workspace W into regions with constant
W
complexity and constant size coverage The rst part is trivial as a rectangloid has
constant complexity the second part is less obvious
The structure of the partition by the supp orting planes of the b ounding b oxes
of the grown obstacles is such that an arbitrary rectangloid region R V either
W
lies in the exterior of all b ounding b oxes in which case it has empty coverage or it
lies entirely in the interior of a number of b ounding b oxes of grown obstacles Let
in the latter case D b e the set of all obstacles E for which R B G E D
B
may have more than a constant number of elements
Let E b e the obstacle in D with the smallest minimal enclosing sphere radius
say We rst prove that no obstacles with minimal sphere radii smaller than
b elong to C ov R Assume for a contradiction that E has minimal enclosing
sphere radius and satis es E C ov R By the de nition of coverage this
means that R G E But then since G E B G E also
B B B
B
R B G E So E must b elong to D violating the assumption that E
is the obstacle in D with the smallest minimal enclosing sphere radius
From E D it follows that R B G E The minimal enclosing hyper
B
sphere radius of E implies that the length of none of the sides of B G E
B
exceeds b b As a result the length of none of the
B
sides of R B G E exceeds b as well An obstacle E with mini
B
mal enclosing sphere radius whose corresp onding grown obstacle G E
B
G
intersects R must itself intersect the enclosing rectangloid R R obtained by
G
growing R by b b in all six axis parallel directions so R R C
B O
B
G
The edges of R have length at most b By Theorem the number
G
of obstacles E with mes radius intersecting the rectangloid region R is
b ounded by a constant and hence the number of grown obstacles G E inter
B
secting R is b ounded by a constant meaning that jC ov R j O
The single algorithmic issue that is to b e solved concerns the computation of the
coverage C ov R E of each region R V b ecause the regions of V and the
W W
adjacencies of E can b e trivially extracted in time n from the the three ordered
W
sequences of planes Instead of taking a single region R V and computing all
W
grown obstacles G E that intersect it we choose a more or less inverse approach
B
here we take a grown obstacle region G E and compute all regions R V that
B W
are intersected by it and add E to all corresp onding sets C ov R under construction
In other words we want to determine all regions R with C ov R E The approach
is to identify a single region R intersected by G E and then use this region as a
B
basis for searching the adjacency graph E to nd the entire connected set of regions
W
intersected by G E The correctness of the approach relies on the connectedness
B
of G E which is implied by the connectedness of E B
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
To nd an arbitrary region R intersected by G E we take a p oint p
B
G E Next we p erform a p oint lo cation query with p p p p in the
B x y z
partition V which by the orthogonality of the partition can b e decomp osed into
W
three binary searches to nd X p Y p and Z p in time O log n The
h x i y j z
Cartesian pro duct X Y Z contains the p oint p and hence intersects G E
h i j B
so we add E to C ov R
In preparation for a second phase we create a set Ev yet consisting of just one
element R We rep eatedly extract an element R from Ev for further pro cessing
until Ev is empty We compute the O neighbors of R and verify for each neighbor
R if R intersects G E If this is the case and the neighbor has not b een treated
B
yet which can b e tested by marking the regions visited then we add E to C ov R
and the neighbor R itself to Ev
The ab ove search through the regions considers a sup erset of the collection of
m regions R V satisfying R G E More precisely it also considers
E W B
all regions adjacent to regions R with R G E Still the total number of
B
regions that are considered is O m As the amount of work p er region is constant
E
the total time sp ent in the search is O m
E
It remains to b ound the number m of regions R V with R G E
E W B
For each region R with R G E we add E to C ov R Hence the sum
B
of all m over all grown obstacles G E equals the sum of all jC ov R j over all
E B
regions R As jC ov R j O for all R V the latter sum amounts to O n
W
As a result the sum of all m equals O n and hence all searches together take
E
O n time The n p oint lo cation queries in the orthogonal cc partition to identify
a starting region for each of the n searches require additional O n log n time in
total As a result the cc partition graph V E and the corresp onding function
W W
C ov can b e computed in n time
Lemma The computation of the cc partition graph V E and the corre
W W
sponding function C ov V P E takes n time
W
The substitution of the computation of the cc partition graph V E and the
W W
function C ov in the rst step of the algorithm FatMot leads by Theorem to a
motion planning algorithm that decomp oses FP into constant complexity sub cells
in time T n n time The number of sub cells and sub cell adjacencies is in the
worst case of the same order of magnitude as the number of regions and adjacencies
in the cc partition
Theorem Let k and b be constants and let E be a collection of k fat
constant complexity obstacles E W IR with minimal enclosing sphere radii at
least Algorithm FatMot solves the motion planning problem for any constant
complexity robot B with f degrees of freedom and reach b amidst E in
B
time n The connectivity graph CG V E of the resulting decomposition
C C
of FP into simple subcells has size O n
SIMILARLY SIZED ARBITRARY OBSTACLES IN SPACE
Similarly sized arbitrary obstacles in space
The motion planning algorithm in the previous section for a rob ot amidst arbitrary
obstacles has a relatively high running time compared to the free space complexity
It is therefore interesting to see what realistic additional assumptions may lead
to a relevant improvement of the p erformance This section shows an interesting
example of such a realistic assumption namely that the obstacles have comparable
sizes The assumption is realistic b ecause in many practical situations the largest
obstacle in the workspace will not b e more than a constant factor bigger than the
smallest obstacle The addition of the assumption to the mildly constrained setting
of the previous section leads to a surprising improvement of the p erformance The
resulting motion planning algorithm computes an optimal O n cell decomp osition
in nearly optimal O n log n time The following problem statement xes the setting
of the results in this section
A constant complexity rob ot B with f degrees of freedom f and
reach moves freely in the workspace W IR amidst a collection E
B
of k fat constant complexity obstacles E W with minimal enclosing
sphere radii in the range u for some constants k and u
The system is constrained by the inequality b for some xed
B
constant b
Again the goal is to compute a small cc partition of the workspace The
b ounded ratio b etween the size of the smallest and largest obstacle in E provides
the opp ortunity of a simple and structured cc partition consisting of axis parallel
rectangloid regions The corners of the rectangloid regions are restricted to the
p oints of the regular orthogonal grid G with resolution More sp eci cally
the rectangloid regions of the sub division are either cub es with side length or
p ossibly semi in nite rectangloids of width and height or rectangloids that are
unbounded in b oth z directions All regions of the latter two types have empty
coverage The number of each of the three types of regions in the sub division is
b ounded by O n Moreover the number of adjacencies is equally low O n The
rectangloid sub division is as such an optimal partition of the workspace
The basic idea b ehind the rectangloid sub division is to embed the b ounding
b oxes B G E of all grown obstacles G E in cub es of the form h h
B B
i i j j where h i j Z The idea seems promising for two
di erent reasons On the one hand the cub es are small enough to certify constant
size coverage while on the other hand the cub es are large enough to b e able to
embed each of the b ounded size grown obstacles in a constant number of cub es
with pair wise disjoint interiors As a result the total number of cub es is linear in
the number obstacles The complement of the cub es clearly has empty intersection
with the grown obstacles The regions of the two non cubic types serve as a means of
e ciently sub dividing this complement All these regions have empty coverage The
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
sub division is such that the regions can b e ordered lexicographically This ordering
on the regions simpli es many of the computations
Let V b e the set of cubic cc partition regions V is de ned by
V E V
E E
where
V E f h h i i j j
jh i j Z h h i i j j B G E g
B
Lemma proves the linear b ound on the number of regions in V
Lemma jV j O n
Pro of Let us b ound the number of cub es in V E for some E E The minimal
enclosing sphere radius of E lies in the range u As a result no two p oints in
E are more than u apart and hence no two p oints in G E are more than
B
u u b apart which in turn implies that the length width and height
B
of the b ounding b ox B G E do not exceed u b Such a b ox is certainly
B
!
embedded in an orthogonal cluster of u b cub es of side length which is
a constant number The number of elements in V E is b ounded by this constant
Summing over all sizes of set V E yields a b ound of O n on the size of the set
V
Lemma con rms the intuition that the cub es of V are so small that they can
only b e intersected by a constant number of grown obstacles
Lemma For al l R V jC ov R j O
Pro of Any obstacle E E for which G E intersects R V must itself
B
intersect the enclosing cub e R of R obtained by growing R by in all three axis
B
parallel directions The resulting R is a cub e with side length b
B
p
and hence with diameter at most b By Theorem the number of
obstacles in E which all have minimal enclosing sphere radi at least intersecting
the cub e R is constant So jC ov R j O
The de nition of V as the union of all sets V E E E gives a clue on a
straightforward but e cient computation of the set V A rst step computes all
constant cardinality sets V E Each set V E of cub es intersecting the grown
obstacle G E is trivially computable in constant time from E and The
B B
resulting sets of cub es are not disjoint a cub e R may o ccur in more than one set
V E To eliminate multiple copies of a single cub e we simply sort the cub es of all
sets V E lexicographically Multiple copies of a single cub e app ear consecutively
in the ordered sequence and are easily ltered out In conclusion the O n log n
time to sort the O n cub es of the sets V E determines the time b ound for the
computation of V
SIMILARLY SIZED ARBITRARY OBSTACLES IN SPACE
Lemma V consists of O n constant complexity regions R with jC ov R j
O the computation of V takes O n log n time
At this stage the remaining task is to nd a way of e ciently partitioning the
!
closure of the complement IR n R into a small number of constant complexity
R V
regions The e cient sub division of the complement outlined b elow pro ceeds in
two phases The rst phase will b e such that up on its completion all workspace
columns of the form h h i i IR that contain a cub e R V are
partitioned into regions with constant complexity and constant size coverage This
approach reduces the remaining step that is the sub division of the complement of
the columns to the essentially two dimensional problem of sub dividing the planar
complement of the intersections of the columns with the plane z Both phases
result in a collection of O n additional regions which are computable in time O n
time using the ordered sequence of cub es from V
The rst of the remaining two steps completes the partition of the union of the
z columns through the cub es of V A z column through a cub e h h i i
j j is the in nite extension h h i i IR of that cub e in the
z direction The complement c n R of the cub es in each z column c through
R V
a cub e of V consists of a collection of p ossibly semi in nite maximal connected
comp onents of height and width see Figure The regions that constitute
the set V are the closures of all such maximal connected comp onents in the O n
"
z columns through cub es of V To understand the contribution to V of a single
"
z column c consider the example of Figure The grey cub es v v in the
#
z column c all b elong to V The complement of the cub es that is the white space
b etween the cub es consists of three connected comp onents w w w of height and
" !
width The closures of these regions form the contribution of the z column c to
V
"
From the construction of the regions of V it is clear that the number of regions
"
!
R V contributed by a z column c can exceed the number of cub es R V in c
"
by at most one as each pair of subsequent regions from V in c must b e separated
"
by at least one cub e from V As a consequence the total number of regions in V
"
is of the same order of magnitude as the number of cub es in V The computation
of the regions of V contributed by a single z column c is simple if the ordered
"
sequence of cub es from V in c is given Fortunately this cub e sequence app ears
as a subsequence in the lexicographical ordering of all cub es of V As a result the
regions of V contributed by all z columns can b e computed by a single scan of the
"
full O n length sequence
Lemma V consists of O n constant complexity regions R with C ov R
"
the computation of V takes O n log n time
"
Note that the computation time of O n log n incorp orates the ordering of the cub es
from V
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
w
"
v
%
v
$
w
!
v
#
v
"
v
!
c
v
w
y
z
x
Figure The column c contains cub es v v all b eing elements of V the
!
regions w w and w are members of V
" # "
#
The problem of nding a partition of the complement IR n R of the
R V !V
!
z columns is a problem that can b e solved in the plane z as such a partition can
b e obtained by orthogonally lifting a planar sub division of the complement of the
square intersections of the columns with z The lifting preserves the asymptotic
complexity of the regions in the partition so it su ces to nd a partition of the
plane into constant complexity regions
Consider the plane z The column cross sections are squares of the form
h h i i with h i Z Figure shows an example of a plane
z with column cross sections To partition the complement of the squares into
constant complexity regions we simply extend vertical walls parallel to the y axis
through the vertical edges of the squares Note that no more than O n walls
are extended partitioning the complement of the squares into at most O n con
stant complexity regions The orthogonal liftings of these regions into space are
constant complexity regions that collectively partition the three dimensional com
plement of the columns
The ordered sequence of cub es of V again turns out a useful to ol in the compu
tation The restriction of the cub es to the rst two co ordinates turns the sequence
into the lexicographical ordering of the squares resulting from the intersection of the
z columns with the plane z Hence the squares app ear from left to right see
Figure and within a vertical slab of the plane from b ottom to top Using this
sequence of squares it is not hard to compute the decomp osition of the plane by
a single scan of the sequence of squares adding an unbounded region like q for #
SIMILARLY SIZED ARBITRARY OBSTACLES IN SPACE
q
#
q
"
q
$
q
q
!
y
x
Figure The regions q q are regions in the vertical decomp osition of the
complement of the column cross sections the grey squares The liftings of these
regions are elements of V
every consecutive pair of squares in non adjacent slabs adding semi in nite regions
like q and q for all rst and last squares in a slab and adding a b ounded region
like q for every consecutive pair of non adjacent squares in a single slab The
sketched computation of V takes O n time provided that the ordered sequence of
cub es from V is given
Lemma V consists of O n constant complexity regions R with C ov R
the computation of V takes O n log n time
The results obtained so far show that the regions of V V V have constant
complexity and constant size coverage In addition the regions collectively partition
IR which makes V V V an adequate choice for a cc partition of the workspace
W IR so we choose V V V V
W
Lemma V V V V is a cc partition of size O n of W with the obstacles
W
E the computation of V takes O n log n time
W
The cc partition of W IR by the regions of V has a recursive structure
W
which turns out to b e useful in the sequel At the upp er level the workspace W
is divided into slices separated by planes x h h Z A slice is either a
region from V like q q in Figure or it is divided into levels by planes y i
i Z and has width A level is either a region from V like q q q or it
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
is a z column of width and height Recall that a z column contains no regions
from V The recursive structure of the sub division shows that all regions in V
W
can b e stored lexicographically that is slice by slice by increasing x level by level
by increasing y within a single slice and by increasing z within a single level The
ab ove structuring makes it easy to distinguish three di erent types of adjacencies
Two adjacent regions are x adjacent if they lie in di erent subsequent slices two
adjacent regions are y adjacent if they lie in di erent subsequent levels of a single
slice two adjacent regions in a single level or z column are z adjacent
We are now ready to prove an upp er b ound on the number of region adjacencies
in the cc partition
Lemma E f R R V V j R R g has size O n
W W W
Pro of The pro of uses case analysis with resp ect to the types of the regions
involved in the adjacency The number of adjacencies in each case is b ounded by
O n
Let us rst count the adjacencies involving a region say R from V A region
R adjacent to the cub e R entirely covers one of the six sides of R regardless of
the type of R Charging the adjcency to the covered side leads to at most O n
chargings and hence at most O n adjacencies involving a region from V
Now consider an adjacency R R V V R is a p ossibly semi in nite
part of a z column and R is unbounded in the z direction R and R must b e
either x adjacent or y adjacent The restriction on the type of adjacency and the
unboundedness of R in the z direction establish that R completely covers one of the
four sides of R parallel to the x z or y z plane Charging the adjacency to the
entirely covered side of R implies that the number of adjacencies R R V V
is O n
The regions involved in an adjacency R R V V are b oth obtained by
lifting a rectangular region in the x y plane orthogonally into the z dimension
Hence the rectangular intersections of R and R with the plane z must b e
adjacent in the planar sub division of z The planar arrangement in z is
a planar graph with O n edges and vertices dividing the plane into O n regions
with a total of O n adjacencies Hence the number of adjacencies R R V V
is O n
The number of the remaining adjacencies R R V V is more di cult to
b ound Both R and R are p ossibly semi in nite parts of di erent z columns R
and R must b e either x adjacent or y adjacent It is easily seen that R and R are
in one of the two relative p ositions depicted in in Figure In the left case one
side involved in the adjacency is contained in the other involved one We charge
the adjacency to the covered side the dashed one in Figure Each side can
only b e charged once The number of sides of regions of V is O n so the number
of adjacencies R R V V of the rst kind is O n In the complimentary
case neither one of the sides is completely contained in the other one In this case
however one edge b ounding the involved side of R is contained in the interior of the
SIMILARLY SIZED ARBITRARY OBSTACLES IN SPACE
y
R
z
R
R R
x
Figure Two types of adjacencies R R V V
involved side of R and vice versa We can charge the adjacency to either edge the
b old edges in Figure Each edge can only b e charged once b ecause it can lie in
the interior of only one face The number of edges of regions of V is O n so the
number of adjacencies of two regions from V of the second kind is O n as well
Combination of all linear b ounds yields that jE j O n
W
For the computation of the set E of adjacencies it is convenient to have
W
the O n regions of V ordered lexicographically Such can b e achieved in time
W
O n log n The computations of the z adjacencies the y adjacencies and the x
adjacencies pro ceed as outlined b elow
The z adjacencies can b e extracted from the sequence of regions in a straight
forward single scan taking O n time Two regions that are z adjacent app ear
consecutively in a z column and also in the lexicographical order Any pair of con
secutive regions in the sequence lying in the same z column should b e rep orted as
an adjacency
At the heart of the computation of the x and y adjacencies lies a basic op eration
that rep orts pairs of adjacent regions in two adjacent levels either in a single slice
or in two subsequent slices Assume that the adjacent levels are divided into m
and n regions resp ectively Then it is easily veri ed that the number of adjacencies
involving one region from either level is m n Moreover the adjacencies can b e
rep orted in time m n by a simultaneous scan of the two levels from z to
z In conclusion the region adjacencies in two adjacent levels can b e rep orted
in time prop ortional to the number of adjacencies
The y adjacencies are restricted to pairs of regions in subsequent levels of a sin
gle slice To identify all pairs of subsequent levels and to compute all adjacencies
induced by their regions we traverse the lexicographical order of regions with two
p ointers The p ointers invariantly p oint to the start of the subsequences corresp ond
ing to two subsequent levels At any combination of p ointer p ositions we apply the
techniques of the previous paragraph to compute the adjacencies induced by the
two levels in time prop ortional to the number of adjacencies After this computa
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
tion b oth p ointers are forwarded to the start of the next level As the number of
y adjacencies is b ounded by O n and b oth p ointers make a single traversal of the
sorted sequence of regions the total time for computing the y adjacencies equals
O n
The computation of the x adjacencies nally resembles the computation of the
y adjacencies Recall that x adjacencies are restricted to pairs of regions in sub
sequent slices Again we traverse the sorted sequence of regions with two p ointers
which now invariantly p oint to the start of comparable height levels in subsequent
slices Comparable height levels in subsequent slices share a common face paral
lel to the y z plane As a result pairs of regions in the comparable levels are
x adjacent The x adjacencies induced by the two levels can b e computed by the
basic strategy outlined ab ove in time prop ortional to the number of adjacencies
After the computation the p ointer corresp onding to the level with the lowest upp er
b oundary with resp ect to the y co ordinate is forwarded to the next level As the
number of x adjacencies is b ounded by O n and b oth p ointers make a single traver
sal of the sorted sequence of regions the total time for computing the x adjacencies
equals O n
To compute the coverage of the regions of V we b orrow the ideas from Section
W
Thus the approach is to consider obstacle by obstacle and compute all regions
R V in fact R V intersected by the corresp onding grown obstacle One
W
arbitrary region R V intersecting the grown obstacle can b e determined in
W
O log n time using the ordered sequence of regions Starting from that region the
other O regions intersected by the grown obstacle are identi ed in O time
using E See Section for the details The approach amounts to O n log n
W
for computing C ov
Lemma The computation of E and C ov takes O n log n time
W
If we substitute the ab ove computation of the cc partition graph V E and the
W W
corresp onding function C ov for the abstract rst step of algorithm FatMot then we
obtain an e cient algorithm for computing a cell decomp osition of the free space
The algorithm yields a decomp osition into O n constant complexity cells in time
T n O n log n by Theorem as T n stands for the time required to compute
V E and C ov
W W
Theorem Let k b and u be constants and let E be a collection
of k fat constant complexity obstacles E W IR with minimal enclosing sphere
radii in the range u Algorithm FatMot solves the motion planning problem for
any constant complexity robot B with f degrees of freedom and reach b
B
amidst E in time O n log n The connectivity graph CG V E of the resulting
C C
decomposition of FP into simple subcells has optimal size O n
PLANAR MOTION AMIDST ARBITRARY OBSTACLES IN SPACE
Planar motion amidst arbitrary obstacles in
space
With the present technological state of the art one rarely encounters free ying
three dimensional rob ots in industrial environments Instead many rob ots move in
three dimensional workspaces amidst spatial obstacles while their motion is con ned
to a planar work oor A realistic example of such a setting is a vacuum cleaner
moving in a ro om in which ob jects hang from the ceiling and stand on the o or
Sometimes the nature and p ositions of the obstacles do es not allow to reduce
such problem to purely planar motion planning The vacuum cleaner for example
can easily pass under a table An approach to solve the problem by pro jecting the
vacuum cleaner and the obstacles onto the o or and then nding a planar path
for the pro jected vacuum cleaner amidst the pro jected obstacles would forbid such
paths In this section we study a general formulation of the type of problem outlined
ab ove in the context of fat obstacles
We consider a workspace W IR with k fat constant complexity obstacles The
motion of the rob ot B in this workspace is constrained by the assumption that a
!
sp eci c p oint p in B is restricted to a work oor F f g IR f g IR the
workspace W is the Cartesian pro duct of the pro jection F of the work oor and the
real line W F IR For convenience we choose the rob ot s reference p oint O B
to b e equal to the p oint p that is restricted to the work oor Hence
O Z F f g
for all placements Z C of the rob ot B The following problem statement xes the
setting of this section
A constant complexity rob ot B with f degrees of freedom f and
reach moves with some p oint O B restricted to a plane F in the
B
workspace W F IR IR amidst a collection E of k fat constant
complexity obstacles E W with minimal enclosing sphere radii at least
for some constant k The system is constrained by the inequality
b for some xed constant b
B
Note that the problem statement do es not restrict the rob ot to rotate around an
axis p erp endicular to the work oor only as for the vacuum cleaner The rob ot is
allowed to rotate arbitrarily and can have more degrees of freedom An example of
such a rob ot is a mo on vehicle equipp ed with several arms to grasp stones etc
Planar motion planning in space is sometimes referred to as two and a half
dimensional motion planning for understandable reasons A solution to this type
of path nding for a p olyhedral rob ot amidst p olyhedral obstacles is given by Wen
tink and Schwarzkopf in Their algorithm which is a generalization of the
b oundary cell decomp osition algorithm by Avnaim Boissonnat and Faverjon
runs in time O n log n Under the realistic assumptions of a b ounded size rob ot
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
and fat obstacles the free space is shown b elow to b e decomp osable into O n
simple sub cells in time O n log n using the general ideas of Section yielding
an O n log n motion planning algorithm
Cho osing the p osition of the rob ot s reference p oint as a part of the sp eci cation
of the rob ot placement similar to the preceding sections seems a bad idea The
fact that the p osition x y z of the rob ot s reference p oint is restricted by z
would have unclear implications for path nding in the resulting con guration space
as not all free placements are valid placements according to the constraint on the
p osition of the rob ot s reference p oint Finding a free path for the rob ot would
require some constrained path nding in such a con guration space Instead we
choose the con guration space to b e
C F D IR D
where D is again some rest space Any p oint Z Z Z x y Z with
F D D
Z F Z D in con guration space corresp onds to a placement of B in which
F D
its reference p oint is p ositioned at x y in the workspace W A p ossible rest space
for a vacuum cleaner would b e D The con guration space C formulated
ab ove makes the application of the tailored paradigm from Section for solving the
motion planning problem imp ossible as the con guration space is not a sup erspace
of the rob ot s workspace
A p ossible strategy for computing a cell decomp osition of the free space would b e
to temp orarily discard the restriction on the p osition of the reference p oint and act
as if the rob ot is free ying and hence has con guration space C W D F
IR D We may b orrow the ideas of Section to decomp ose the free part FP of C
into O n simple sub cells in O n log n time The free part FP of the con guration
space C F D of the constrained problem can b e regarded as the pro jection onto
the space F D of the subset FP F f g D of the free part FP C W D
To obtain a cell decomp osition of the free part FP C F D we simply
intersect all sub cells and common b oundaries in the decomp osition of FP C
W D with the space F f g D and subsequently pro ject the intersection onto
the subspace F D Regardless of the p ossibly unnecessarily large complexity
of the resulting decomp osition the computation takes O n log n time which is
inferior to the approach that we follow b elow leading to a decomp osition of size O n
in time O n log n The approach exploits the general paradigm for cylindri able
con guration spaces which is given in Section as an algorithm for transforming
a base partition of some appropriate base space into a cell decomp osition of the free
space
The pro jected work oor F turns out to b e a go o d choice for a base space for the
cylindrical decomp osition of the free space The sub division that we prop ose strongly
resembles the partition of the workspace W IR by the grown obstacles discussed in
Section Consider the planar arrangement in F f g de ned by the intersection of
the grown obstacle b oundaries G E W and the work oor F f g It will b e B
PLANAR MOTION AMIDST ARBITRARY OBSTACLES IN SPACE
shown that this arrangement partitions the plane F into regions whose corresp onding
con guration space cylinders are intersected by a constant number of constraint
hypersurfaces The O n curves resulting from the intersection of grown obstacle
faces with F f g have constant complexity due to the constant complexity of the
grown obstacles The arrangement of curves has complexity O n To obtain a valid
base partition in the base space F we compute inspired by resemblance with Section
the vertical decomp osition of the planar arrangement of intersection curves in
time O n log n time The resulting base partition graph has complexity O n
The computation of the constraint hypersurfaces that intersect the con guration
space cylinders is supp orted by sets that resemble the region coverages and are
computed along with the vertical decomp osition in a way that strongly resembles
the computation of the coverages Below we settle the details of the informally
describ ed approach
A restriction on the applicability of the general paradigm is that the con guration
space C is cylindri able that is decomp osable in some B and D such that for all
p B
jf f j B E f p D gj O
f f
Lemma states that the condition is satis ed for B W but W is not a subspace
of C Fortunately the prop erty is inherited by the subset F f g W leading to
the following prop erty
Prop erty For al l p F
jf f j B E f p D gj O
f f
As a result the two dimensional Euclidean space F is a feasible base space for a
cylindrical decomp osition of FP C F D The algorithm in Section
transforms the graph V E corresp onding to a base partition in F into a cell
F F
decomp osition of the free part of C provided that all regions R V have constant
F
complexity and satisfy
jf f j B E f R D gj O
f f
In the previous chapter we have formalized the informal observation that a rob ot
can only touch an obstacle lying within its reach using the notion of grown obstacles
A rob ot with its reference p oint at some p oint p W is only able to touch an obstacle
E if p G E In our constrained case the p osition p of B s reference p oint for
B
p otential collision with E E is further restricted to p G E p F f g
B
In other words the reference p oint of B must b e p ositioned at some p oint in the
intersection of the grown obstacle G E and the plane F f g IR f g Notice
B
that the emptiness of the intersection implies that B cannot collide with E during its
constrained motion We de ne G E to b e the intersection G E F f g
F B B
Actually it is the intersection restricted to the rst two co ordinates to make it lie in F
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
De nition G E f x y j x y G E g
F B B
The restriction on the p osition of the reference p oint of the rob ot B for b eing able
to touch an obstacle E has implications for the lo cation in con guration space of
the constraint hypersurfaces de ned by a contact of a feature of B with a feature of
E
Lemma Let B and E Then
f f
f G E D
F B
Pro of Let p p p p p p f such that p F and p D We
F D x y D F D
must prove that p p p p G E D which may b e reduced to proving
x y D F B
that p p G E as p D is trivially true Proving p p G E is
x y F B D x y F B
in turn equivalent to proving p p G E This means that B s reference
x y B
p oint must b e placed inside G E when B s feature touches
B
Assume for a contradiction that p p G E Then by the de nition
x y B
of grown obstacles the distance from p p to E exceeds But then it is
x y B
imp ossible for B to reach and touch the obstacle E by the de nition of the reach
of a rob ot In other words no feature B can touch a feature E So
f f
the p oint p p p p p p with p p G E cannot lie on f
F D x y D x y B
contradicting the assumption of the lemma
Lemma provides similar to Lemma in Section for free ying rob ots
some simple outer approximation on the lo cation of any constraint hypersurface
f in the con guration space C F D If R F do es not intersect G E
F B
then no constraint hypersurface f with E intersects the con guration space
f
cylinder G E The following de nition simpli es a more general formulation
F B
of the consequences of Lemma
De nition Let R F IR
C ov R fE E j R G E g
F F B
C ov R is the collection of obstacles E whose corresp onding regions G E
F F B
intersect R Now if a region R F is intersected by a collection of regions G E
F B
then the con guration space cylinder R D can only b e intersected by constraint
hypersurfaces induced by the corresp onding obstacles E or more formally
ff j B E f R D g
f f
ff j B C ov R g
f f F
The set inclusion directly shows that if R is chosen such that jC ov R j O
F
then jf f j B E f R D gj O due to the constant
f f
PLANAR MOTION AMIDST ARBITRARY OBSTACLES IN SPACE
complexities of the obstacles and the rob ot Hence any partition of F into constant
complexity regions R with jC ov R j O is a valid base partition for a cylindrical
F
cell decomp osition of FP C and as such a valid input to the transformation
algorithm Let us now fo cus on the arrangement of regions G E in the plane
F B
F
The planar arrangement A G of all b oundaries G E E E sub divides
F F B
F into maximal connected sets of p oints p F with equivalent collections C ov p
F
Lemma states that the arrangement has complexity O n In addition it states
that each cell A in A G satis es jC ov A j O F f g
F F
Lemma Let A G be the planar arrangement of al l boundaries G E
F F B
E E Then
a A G has complexity O n
F
b jC ov A j O for al l faces A A G
F F
Pro of Theorem yields for the spatial arrangement A G of all grown ob
stacles G E E E i the complexity of A G is O n and ii every p oint
B
p W IR lies in at most O regions G E E E The intersection of
B
the linear complexity arrangement A G with the plane F f g results in a planar
arrangement in F f g with complexity O n Hence A G has complexity O n
F
To prove b it su ces to pick a p oint p A and prove that it lies in at most O
regions G E b ecause all p oints in a single face A lie in exactly the same re
F B
gions By expression ii every p oint p F f g W lies in at most O regions
G E F f g yielding b
B
The arrangement A G partitions the base space F into regions whose cor
F
resp onding con guration space cylinders are intersected by only O constraint
hypersurfaces The regions however do not have constant complexity but thanks
to the planarity of the sub division such is easily remedied by vertical decomp osition
of the arrangement The resulting vertical decomp osition regions are the regions of
V The set E f R R V V j R R g consists of all adjcencies of pairs
F F F F
of regions Section explains why the vertical decomp osition of an arrangement
of complexity O n consists of O n regions and region adjacencies
Lemma V consists of O n constant complexity regions R that partition F
F
and satisfy jC ov R j O E has size O n
F F
The transformation algorithm in Section transforms the base partition rep
resented by the sets V and E into a decomp osition of the free space in time
F F
O jV j jE j provided that for every region R the collection of constraint hyper
F F
surfaces ff j B E f R D g is computable in constant time
f f
Fortunately we have found that ff j B E f R D g f f j
f f f
B C ov R g O which shows that a constant time computation of
f F
CHAPTER EFFICIENTLY COMPUTABLE BASE PARTITIONS
the appropriate hypersurfaces is p ossible when the function C ov V P E is
F F
given Therefore we decide to compute the function C ov along with the vertical
F
decomp osition The similarity of the present triple V E C ov and the triple
F F F
V E C ov in Section suggests the use of the plane sweep from that section
W W
for the simultaneous computation of V E and C ov in O n log n time The
F F F
input to the sweep are the O n lab eled maximal connected x monotone arcs of
the b oundaries G E E E having no vertices in their interiors The arcs
F B
of a single b oundary G E are obtained in constant time by intersecting the
F B
constant complexity grown obstacle G E with the plane F f g and subse
B
quently cutting the pro jection of the intersection into the appropriate x monotone
arcs The generation of all input arcs takes O n time so the entire computation of
the base partition V E and the function C ov from the grown obstacles takes
F F F
O n log n time
Lemma The computation of the base partition V the set E and the function
F F
C ov takes O n log n time
F
The computation of a cell decomp osition consists of two steps the rst step
which computes a valid base partition and the second step which transforms the base
partition into a decomp osition of the free space The second step takes time O jV j
F
jE j O n b ecause the function C ov supp orts the constant time computation
F F
of ff j B E f R D g for any region R The combination with
f f
the O n log n running time for the rst step justi es the following theorem which
formulates the main result of this section
Theorem Let k and b be constants and let E be a collection of k fat
constant complexity obstacles E W IR with minimal enclosing sphere radii at
least Algorithm FatMot solves the motion planning problem for any constant
complexity robot B with f degrees of freedom and reach b amidst E
B
whose reference point O B is con ned to the planar work oor F f g in time
O n log n The connectivity graph CG V E of the resulting decomposition of
C C
FP into simple subcells has optimal size O n
Chapter
Concluding remarks
This nal chapter recaptures some of the main results in this thesis and gives some
re ections on p ossible extensions and improvements The reader is warned that
most re ections are based on intuitive feelings and not supp orted by indisputable
pro ofs
We have studied the motion planning problem for a constant complexity rob ot
d
B with f degrees of freedom amidst n constant complexity k fat obstacles E IR
for some constants d f and k In addition the reach of the rob ot B
B
is assumed to b e b ounded from ab ove by a constant multiple b of where
is a lower b ound on the minimal enclosing hypersphere radius of any obstacle
E The mild assumptions are considered to provide a realistic framework for many
practical motion planning problems The complexity of the free space for problems
that satisfy the assumptions was proven to b e O n whereas the complexity can
f
easily b e as high as n when b oth the fatness assumption on the obstacles and
the b ounded size assumption on the rob ot are dropp ed This remarkable gap makes
it interesting to study the individual in uence on the free space complexity of each
of the two assumptions in more detail which we leave as an op en question
The basis of the linear complexity result is a low obstacle density prop erty rela
d
tive to the rob ot size of the workspace W IR which is implied by the combination
of the two assumptions As such the low obstacle density prop erty imp oses a weaker
restriction for obtaining linear complexity free spaces To see that the latter condi
tion is weaker imagine a workspace with non fat obstacles that are far apart All
algorithmic motion planning results in this thesis however apply equally well to
problems that satisfy the relaxed condition
Besides having a low combinatorial complexity the free space for a motion plan
ning problem that ts in our framework also has a b ene cial structure The structure
allows for a decomp osition of the con guration space into cylinders with bases in
some pro jective subspace the so called base space such that the free space part of
every cylinder has constant complexity In other words the cylinder walls partition
the free space into constant complexity parts The maximal connected comp onents
of the free parts make p erfect sub cells in a cell decomp osition of the free space as
CHAPTER CONCLUDING REMARKS
they allow for simple path nding in their interiors due to their constant complexity
and connectedness The validity of a pro jective subspace as a base space is veri ed
by a pro of that the lifting back into con guration space of every p oint in the base
space is intersected by at most a constant number of constraint hypersurfaces which
are sets of contact placements of a rob ot and an obstacle feature The preceding
considerations reduce the problem of nding a cell decomp osition of the free space
to the problem of nding some constrained decomp osition of the lower dimensional
base space in which the regions are appropriate cylinder bases A uniform sequence
of op erations then su ces to transform the base partition into a cell decomp osi
tion of the free space of asymptotically equal size The running time of the entire
paradigm is determined by the time to compute the base partition A small and ef
ciently computable base partition is therefore of obvious imp ortance to the success
of the approach Finding a small and e ciently computable base partition may seem
a hard problem at rst sight b ecause the regions are constrained by a restriction
on their liftings hence in con guration space
The extensive and interesting class of motion planning problems with con gu
ration spaces C W D which includes for example all problems that involve
a free ying rob ot allows for choosing the rob ot s workspace W as a base space
Moreover any so called cc partition of W which is sub ject to constraints that are
formulated exclusively in W turns out to b e a valid base partition of the base space
W A cc partition decomp oses the workspace into constant complexity regions that
intersect at most a constant number of cells of the arrangement of grown obstacle
b oundaries A grown obstacle is the collection of p oints within a distance from
B
the original obstacle The arrangement of grown obstacle b oundaries has complex
ity O n and each p oint p ! W lies in no more than a constant number of grown
obstacles simultaneously
Optimal O n size cc partitions exist for three out of ve practical instances
of the motion planning problem amidst fat obstacles studied in this thesis These
instances are motion planning in the plane amidst arbitrarily shap ed obstacles mo
tion planning in space amidst arbitrarily shap ed obstacles of comparable sizes and
motion planning on a work oor in space amidst arbitrarily shap ed obstacles The
reason for treating restricted instances of the motion planning problem in space
lies in the failure to nd an optimal partition for the general version of the problem
and in the frequent o ccurrence of these sp eci c instances in real life motion planning
problems All three cc partitions are computable in nearly optimal O n log n time
In conclusion we have obtained O n log n motion planning algorithms for each of
the three classes of problems The algorithms yield a cell decomp osition of optimal
size O n Notice that these results do not dep end on the number of degrees of
freedom of the rob ot
Note that this description of a cc partition seems somewhat di erent from the formal de nition
in Chapter Nevertheless it is essentially equivalent due to the constant size coverage of the arrangement cells
The cc partitions that are obtained for the two remaining instances motion plan
ning in space amidst p olyhedral obstacles and amidst arbitrarily shap ed obstacles
! !
have sizes O n and O n and are computable in time O n log n and n re
sp ectively The partitions give rise to motion planning algorithms with running
!
times O n log n and n that compute cell decomp ositions of sizes O n and
!
O n for the resp ective problems regardless of the number of degrees of freedom of
the rob ot These results might b e improvable The challenge is to nd sub quadratic
and sub cubic partitions of the workspace W such that each region has constant com
plexity and intersects no more than a constant number of cells of the arrangement
of grown obstacle b oundaries
Besides attempting to improve the latter two results it is also interesting to see
if the paradigm for motion planning amidst fat obstacles applies to other classes
of motion planning problems involving fat obstacles Let us give some thoughts
on some p ossible extensions like motion planning with moving obstacles multiple
rob ots and anchored rob ot arms
Motion planning problems involving moving obstacles are normally solved in
con guration time space The con guration time space C T is the Cartesian pro d
uct of the con guration space C of the stationary version of the problem and the
time dimension T The fact that motion back in time is imp ossible is re ected by the
additional requirement that any solution curve b etween a pair of query placements
in the con guration time space must b e strictly monotone in time The requirement
imp oses restrictions on the search of the connectivity graph of a cell decomp osition of
the free part of C T and on the simple motions within each sub cell of the decomp osi
tion The complexity of the free part of the con guration time space C T can increase
rapidly when many obstacles are non stationary and travel along complicated tra
jectories The ideas of the preceding two chapters will de nitely not b e applicable
in such cases Things seem di erent when we assume that only a constant number
c out of the n obstacles move along simple tra jectories that is algebraic curves of
b ounded degree Let us assume furthermore that a cc partition of the d dimensional
workspace with the n c stationary obstacles is given Now the cylinders obtained
by lifting the d dimensional cc partition into the f dimensional con guration
time space are intersected by only a constant number of constraint hypersurfaces
de ned by contacts of the rob ot and the stationary obstacles The overall number
of constraint hypersurfaces due to contacts of the rob ot and the moving obstacles
is constant The simplicity of the motion supp osedly implies that the intersection
of each such constraint hypersurface with a cylinder consists of a constant number
of constant complexity connected comp onents Hence the constraint hypersurfaces
de ne constant complexity arrangements in every cylinder Therefore the results of
the previous chapter seem to generalize directly to environments in which a constant
number of the obstacles are non stationary When the number of moving obstacles
is not constant the preceding arguments no longer hold It is though to b e exp ected
that still a large complexity reduction is achievable from the fatness of the mov
ing and stationary obstacles it follows that any cross section of the free part of C T
CHAPTER CONCLUDING REMARKS
at a particular time t has linear complexity
The usual approach to the exact solution of a motion planning problem with
c rob ots B B with con guration spaces C C of dimensions f f is
c c c
centralized planning In centralized planning the c rob ots are regarded as one multi
b o dy rob ot B B B Planning the motion of the multi bo dy rob ot B takes
c
place in the comp osite con guration space C C C There collisions of the
c
rob ots B i c turn into collisions of the multi bo dy rob ot B The alternative
i
to centralized planning decoupled planning plans the motions of each of the rob ots
indep endently and then considers the interactions of the resulting paths Decoupled
planning is not guaranteed to nd a solution to the problem The complexity of
the free part of the comp osite con guration space C can for fat obstacles easily b e
c
as high as n even when the reaches of the individual b o dies B i c
B i
i
are b ounded by b for some constant b The key observation here is
B
i
that the reach of the comp osite rob ot is in no way b ounded two b o dies B and
B i
B can b e in nitely far apart Figure shows a single c fold contact of B Clearly
j
c c
there are n such contacts Still there is a considerable gap b etween the n
E E E E E
! " n n
B B B
k i j
Figure Each of the c rob ots B B touches one of the n obstacles E E
c n
This corresp onds to a single c fold contact of the comp osite rob ot B B B The
c
c
total number of such contacts is n
P
f
lowerbound construction and the obvious upp erb ound of O n with f f
i
i c
on the complexity of the free space We b elieve the complexity of the free space to
b e close to the lowerbound The ideas of cylindrical decomp osition of the free space
d
seem applicable to some extent if the workspace W IR is a pro jective subspace
of each of the con guration spaces C i c Then the comp osite workspace
i
c c
W is a pro jective subspace of the comp osite con guration space C A p oint p W
xes the p ositions of the reference p oints of all b o dies B The low obstacle density
i
of the workspace and the b ounds on the sizes of the individual b o dies yield that
each b o dy can touch only a constant number of obstacles while its reference p oint
is xed Provided that c is a constant the lifting of p into C is intersected by O
constraint hypersurfaces Hence C is a cylindri able con guration space and W is a
valid base space The existence of a small and e ciently computable base partition
remains an op en question
The straightforward application of the framework of assumptions in the second
paragraph of this chapter to an anchored rob ot arm do es not give rise to an inter
esting motion planning problem Even though the number of obstacles can b e high
the total number of obstacles touched by the rob ot in any placement is at most con
stant due to its b ounded size The free space of the anchored rob ot has constant
complexity and a rigorous cell decomp osition metho d su ces to compute a cell
decomp osition of the free space in constant time
A more interesting problem formulation follows when we take a closer lo ok at
industrial rob ot arms Typically the links close to the base of the arm are long
major axes whereas the links close to the tip or hand are short minor axes
Figure shows a rob ot arm with two ma jor axes L and L and two minor axes
!
L and L Now consider an f link rob ot arm B of which the m links closest to the
" #
L
!
L
"
L
#
L
Figure A rob ot arm with two ma jor axes L and L and two minor axes L and
! "
L
#
tip are not to o large compared to the obstacles The sizes of the f m ma jor axes
are not b ounded Assume that C is the f m dimensional con guration space
corresp onding to the ma jor axes and that C is the m dimensional con guration
!
space corresp onding to the minor axes Hence C C C is the f dimensional
!
con guration space of B A p oint p C xes the placements of all ma jor axes If m
is a constant then the m minor axes can only touch a constant number of obstacles
while the ma jor axes are xed due to the low obstacle density As a result the
lifting of the p oint p C into C will b e intersected by only a constant number of
constraint hypersurfaces So the con guration space C is cylindri able and C is a
valid base space Again however the existence of a small and e ciently computable
base partition in C remains uncertain Nevertheless we exp ect the complexity of
f m f
the free space to b e closer to O n then to the obvious upp erb ound of O n
We can conclude that the paradigm presented in this thesis might lead to many
more e cient motion planning algorithms for a variety of instances of the problem
!" CHAPTER CONCLUDING REMARKS
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Bibliography
Chapter introduces a universal notion of fatness A preliminary abstract on the
complexity of the free space for motion planning amidst fat obstacles used a
di erent notion of fatness which is introduced as thickness in Chapter The
failure to prove a low free space complexity for motion planning problems involving
non convex thick obstacles has led to the current slightly more restrictive fatness
notion
Chapter generalizes earlier rep orted results on p oint lo cation and range search
ing among sets of fat ob jects in and space to spaces of arbitrary dimension d
by applying di erent to ols The general results are rep orted in
The results of Chapter on the complexity of the free space for a rob ot moving
amidst fat obstacles app eared in
The O n log n time b ound on the running time of a fat version of the Piano Movers
algorithm in Section improves the earlier rep orted b ound of O n log n in
b oth and The basis of the gain lies in a more e cient way of searching for
pairs of neighboring obstacle features through a plane sweep of enveloped features
rather than by orthogonal range searching or windowing among obstacle edges
An extended abstract of the contents of Chapters and can b e found in A
full version of the pap er is in preparation !"
Index
cylindri able Ackermann function
inverse dimension
admissible p osition
con guration space obstacle
algebraic decomp osition con guration time space
arm
connected comp onents
arrangement
connectivity graph
cell
edge
j face
vertex
axis
constrained cylindri cation
ma jor
constraint hypersurface
minor
self collision
coverage
base
critical curve
base partition
section
base space
critical orientation
binary space partition
binary tree
Davenport Schinzel sequences
balanced
decoupled planning
b ounded lo cal complexity
degrees of freedom DOF
b ounding b ox
depth order
divide and conquer
cc partition
graph
ellipsoid
cell
volume
complexity
envelope
cell decomp osition metho d
upp er lower
approximate
event point schedule
b oundary
fat sub division
exact
fat
centralized planning
k fat
closed
FatMot
closure
fatness
collision
lack of computational geometry
feature con guration
forbidden cell con guration space
INDEX
move fractional cascading
dynamic
multiple contacts
multiple rob ots free cell
free space FP
non holonomic constraint
b oundary BFP
noncritical region
complexity of the
obstacle
general p osition
non stationary
grown obstacle
obstacle feature
hand
op en
hidden surface removal
paradigm
hypercub e
path
hypersphere
collision free
volume
physical space
hypersphere volume multiplier
Piano Movers algorithm
incremental construction
placement
interior
contact
forbidden
joint
free
prismatic
semi free
revolute
p oint lo cation problem
p otential eld metho d
k fat
priority queue
ladder
quad edge structure
link
list
range searching problem
low obstacle density
range tree
layered
mes radius
reach
minimal enclosing hypersphere
reference p oint
minimal enclosing hypersphere radius
regular orthogonal grid
resolution
Minkowski di erence
retraction
molecule mo del
retraction function
motion
retraction metho d
collision free
Voronoi
motion planning metho d
b oundary vertices
approximate
rigid exact
roadmap
motion planning problem
roadmap metho d
rob ot general
INDEX
articulated
free ying
rob ot arm
rob otics
segment tree
multi level
self collision
semi free space SFP
single cell
slab
sparsity
sub cell
sweep
sweep line status
k thick
thinness
tip
union b oundary complexity
vacuum cleaner
vertical decomp osition
full
planar
three dimensional
Voronoi diagram
wall
wedges
double
wide
work oor
workspace
wrapping
!" INDEX
Acknowledgements
I thank all p eople that have contributed directly or indirectly in making this thesis
into what it is In particular I wish to thank
my sup ervisor Mark Overmars he has b een available for inspiring discussions at
almost any moment it has b een a pleasure working for and with him
the present and former members of the Vakgroep Informatica at Utrecht University
and particularly those in the eld of computational geometry for creating a pleasant
research environment
the p eople at the School of Mathematical Sciences at Tel Aviv University and es
p ecially Dan Halp erin currently at Stanford University for making my stay in Tel
Aviv in March and April b oth instructive and enjoyable
Marko de Gro ot and Maarten Pennings for b eing pleasant ro ommates during di er
ent p erio ds of my stay at the Vakgroep Informatica
the members of the reviewing committee Prof J van Leeuwen Utrecht University
Prof F Gro en University of Amsterdam Prof C Erkelens Utrecht University
and Dr J D Boissonnat INRIA Sophia Antipolis France for reviewing this
thesis
my parents for supp ort and for their interest in my work
and nally Petra for supp ort and care sp ending almost hours a day together
turned out to b e so easy !"
Samenvatting
Het gro eiende aantal to epassingsgebieden voor rob ots en de to enemende verschei
denheid in hun taken vereist steeds meer autonomie van de rob ots Een autonome
rob ot accepteert complexe taken en voert die uit zonder hulp van zijn omgeving
Een voor de hand liggende op dracht voor zo n autonome rob ot is om van een b egin
p ositie naar een eindp ositie te b ewegen waarbij b otsing met de aanwezige obstakels
vermeden dient te worden Het vinden van een dergelijk pad wordt het motion
planning probleem geno emd
Het motion planning probleem wordt in het algemeen opgelost in de con gu
ratieruimte Dit is de ruimte van de representaties van alle mogelijke rob otp osities
ofwel con guraties Het aantal vrijheidsgraden van de rob ot b epaalt de dimensie van
de con guratieruimte Een con guratie is vrij wanneer de rob ot in de overeenkom
stige p ositie geen enkel obstakel do orsnijdt Indien de rob ot in een p ositie een of
meerdere obstakels do orsnijdt dan is de b etre ende con guratie verboden De vrije
ruimte FP is de deelruimte van de con guratieruimte die b estaat uit alle vrije con
guraties Het oplossen van het motion planning probleem in de con guratieruimte
komt neer op het vinden van een continue curve die de b egincon guratie met de
eindcon guratie verbindt en b ovendien volledig is b evat in de vrije ruimte De con
tinue curve in de con guratieruimte komt overeen met een b otsingsvrij pad voor de
rob ot in zijn werkruimte Het is tamelijk eenvoudig om in te zien dat de mo eilijkheid
van het vinden van een continue curve in de vrije ruimte ofwel het oplossen van het
motion planning probleem sterk afhankelijk is van de complexiteit b eschrijvingsg
ro otte van die ruimte Op haar b eurt hangt de complexiteit van de vrije ruimte in
hoge mate af van het aantal meervoudige contacten van de rob ot met de obstakels
Helaas kan in theorie het aantal meervoudige contacten en dus de complexiteit van
de vrije ruimte erg ho og zijn
Motion planning algoritmen trachten een continue curve in de vrije ruimte te
vinden De vrije ruimte is daarbij indirect gegeven do or middel van de rob ot en
de obstakels Aangezien zo n continue curve do or het complexe karakter van de
vrije ruimte zelfs voor eenvoudige motion planning problemen onmogelijk direct te
b epalen is is verdere verwerking van de vrije ruimte no o dzakelijk De verschillende
motion planning algoritmen onderscheiden zich do or de manier waarop zij de vrije
ruimte verwerken tot een structuur waarmee men in staat is om op e ci ente wijze
een pad te vinden tussen twee con guraties
SAMENVATTING
In de huidige praktijk worden vrijwel altijd benaderende methoden gebruikt om
het motion planning probleem op te lossen Benaderende metho den verwerken de
vrije ruimte tot een structuur die de vrije ruimte b enadert Ze vinden in veel gevallen
snel een pad voor de rob ot do ch in sommige lastige gevallen zullen ze er niet in slagen
om een oplossing te vinden Sinds een aantal jaren b estaan er o ok een aantal exacte
methoden Deze metho den die hun o orsprong voornamelijk in de computationele
geometrie gemeenschap hebb en vinden gegarandeerd een pad wanneer er een pad
b estaat Dit pro efschrift concentreert zich op exacte metho den
Exacte metho den kunnen op grond van hun aanpak grofweg in twee categorie en
worden ingedeeld cel decompositiemethoden en retractiemethoden Cel decomp osi
tiemetho den verdelen de vrije ruimte in eenvoudige subcellen Deze sub cellen vormen
de knop en van een graaf Twee knop en van de graaf zijn met elkaar verbonden
wanneer de overeenkomstige sub cellen aan elkaar grenzen De aanpak reduceert
het motion planning probleem tot het vinden van een pad in de graaf Do or de
vereiste eenvoud van de sub cellen is het aantal b eno digde sub cellen om de vrije
ruimte te verdelen in hoge mate afhankelijk van de complexiteit van de ruimte
Retractiemetho den trachten de structuur van de vrije ruimte vast te leggen in een
een dimensional netwerk van curves in diezelfde ruimte het wegennet Dit houdt
in dat elke vrije con guratie via een eenvoudig pad verbonden dient te zijn met
het wegennet en dat alle curves in een samenhangende comp onent van de vrije
ruimte met elkaar een samenhangende comp onent van het wegennet vormen Deze
voorwaarden reduceren het motion planning probleem wederom tot het vinden van
een pad in een graaf namelijk het wegennet De b eide eisen aan het wegennet
maken de gro otte van het wegennet sterk afhankelijk van de complexiteit van de
vrije ruimte Uiteraard b e nvloedt de gro otte van de b erekende structuur graaf de
rekentijd van de exacte algoritmen
De conclusie uit het voorgaande is dat de e ci entie van exacte algoritmen sterk
afhangt van de complexiteit van de vrije ruimte Aangezien die complexiteit in
theorie erg ho og kan zijn lijken exacte metho den ongeschikt voor to epassing in
praktijksituaties De literatuur to ont echter dat de omstandigheden dat wil zeggen
vorm en p osities van de obstakels en vorm van de rob ot die leiden tot de hoge
complexiteiten vaak een kunstmatig karakter hebb en en vrijwel no oit voorkomen in
praktische motion planning problemen In praktijkgevallen zal de complexiteit van
de vrije ruimte ver b eneden de theoretische grenzen blijven Voor dergelijke gevallen
zal de to epassing van exacte algoritmen dan wellicht realistisch worden Een studie
naar milde voorwaarden die een b ewijsbaar lage complexiteit van de vrije ruimte
tot gevolg hebb en is daarom van gro ot b elang voor de to epasbaarheid van exacte
metho den
Dit pro efschrift to ont dat de combinatie van vette obstakels en een niet al te
grote rob ot leidt tot een drastische reductie van de complexiteit van de vrije ruimte
Een vet obstakel is een obstakel dat niet lang en dun is en o ok niet dergelijke delen
heeft Vetheid vormt in een aantal problemen uit de computationele geometrie een
realistische aanname die resulteert in lage complexiteiten en e ci ente algoritmen
SAMENVATTING
Veel praktische motion planning problemen combineren een niet te grote rob ot met
een zekere vetheid van de aanwezige obstakels zo dat vetheid o ok in motion planning
een waardevol b egrip is
De b ovengenoemde omstandigheden leiden tot een verlaging van de rekentijd van
een aantal b estaande cel decomp ositie en retractiemetho den De kern van de ef
ci entiewinst voor deze metho den ligt in een lage obstakeldichtheid in de werkruimte
die een direct gevolg is van de vetheid van de obstakels De eigenschap vormt de basis
voor een nieuwe algemene aanpak of paradigma voor motion planning problemen
tussen vette obstakels Het paradigma volgt de cel decomp ositie aanpak
Het gepresenteerde paradigma voor motion planning tussen vette obstakels is
to epasbaar op problemen waarvoor de werkruimte een pro jectieve deelruimte is van
de con guratieruimte Wanneer de rob ot vrij b eweegt in de werkruimte en dus
niet is verankerd dan zal in het algemeen aan deze voorwaarde voldaan zijn De
aanpak reduceert het probleem van het vinden van een cel decomp ositie van de vrije
ruimte tot het probleem van het vinden van een verdeling van de werkruimte met de
vette obstakels die aan b epaalde voorwaarden voldoet Een aantal uniforme stapp en
b erekenen vervolgens een cel decomp ositie van de vrije ruimte uit de verdeling van
de werkruimte Het aantal sub cellen is direct afhankelijk van de gro otte van de
werkruimte verdeling Optimale verdelingen blijken te b estaan voor motion planning
problemen in het vlak voor motion planning in een drie dimensionale ruimte met
obstakels van vergelijkbare gro otte en voor motion planning op een werkvloer in
een drie dimensionale werkruimte met obstakels De inpassing van de verdeling en
de b erekening ervan in het paradigma resulteert in zeer e ci ente algoritmen voor de
b etre ende problemen Go ede verdelingen en dus e ci ente algoritmen b estaan o ok
voor motion planning problemen in een drie dimensionale ruimte met p olyhedrale en
willekeurige obstakels van onbep erkte afmetingen Verbeteringen van deze laatste
resultaten lijken echter mogelijk De e ci entie van elk algoritme dat volgt uit het
paradigma is onafhankelijk van het aantal vrijheidsgraden van de rob ot ofwel de
dimensie van de con guratieruimte in tegenstelling tot de meeste andere exacte
motion planning algoritmen
Naast motion planning b esteedt het pro efschrift o ok aandacht aan de rol van
vetheid in twee kernproblemen in de computationele geometrie point location en
range searching Het p oint lo cation probleem vraagt om gegeven een aantal niet
snijdende ob jecten voor een willekeurig punt te rapp orteren welk ob ject het punt
b evat of om te concluderen dat geen enkel ob ject het punt b evat Het range searching
probleem vraagt om voor een willekeurige regio de verzameling do orsneden ob jecten
te rapp orteren Het pro efschrift laat zien dat wanneer de ob jecten vet zijn e en
enkele datastructuur volstaat om b eide problemen e ci ent op te lossen De oplossing werkt in willekeurige dimensie
!" SAMENVATTING
Curriculum Vitae
Arnoldus Franciscus van der Stapp en
oktob er geb oren te Eindhoven
VWO aan het Bisschop Bekkerscollege te Eindhoven
studie Informatica aan de Technische Universiteit Eindhoven
afstudeerverslag Distributed data structures for sparse linear algebra
assistent in opleiding in de p ost do ctorale ontwerpersopleiding
Technische Informatica aan het Instituut Vervolgopleidingen van de
Technische Universiteit Eindhoven
pro jectverslag Planning and scheduling in crude oil re neries
ISBN X
onderzo eker in opleiding aan de Vakgroep Informatica van de
Universiteit Utrecht in dienst van de Nederlandse Organisatie voor
Wetenschappelijk Onderzo ek N W O
to egevoegd onderzo eker aan de Vakgroep Wiskunde van de
Universiteit Utrecht