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Motion Planning Amidst Fat Obstacles

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Motion Planning Amidst Fat Obstacles 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 Magnicus Prof dr JA 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 MH Overmars Faculteit Wiskunde en Informatica CIPGEGEVENS 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 entic Research NWO 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 dened 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 Boundaryvertices retraction Fatnesssensitive cell decomp osition Complexity of the cell decomp osition Computing the cell decomp osition A p olygonal rob ot i ii CONTENTS A fatnessinsensitive 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 freeying rob ots Eciently computable base partitions Arbitrary obstacles in space Polyhedral obstacles in space Arbitrary obstacles in space Similarlysized 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 eectively replace humanbeings 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 highlevel 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 collisionfree path is the sub ject of this thesis A rob ot is a movable mechanical device op erating in a physical world the rob ots 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 dicult 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 freeying In this thesis we will mainly deal with freeying 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 freeying articulated rob ot in the plane consisting of two links ! and B is a freeying rigid rob ot in threedimensional space " in a twodimensional 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 dene 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 twodimensional 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 suces to uniquely sp ecify a " placement of B if the link L is allowed to rotate as well Then the rob ot can take " innitely 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 xaxis solves the problem " " Any quadruple x y IR sp ecies 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 threedimensional workspace is a socalled rigid rob ot consisting of # # one solid nondeformable 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 suces 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 innitely many dierent 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 conguration of the rob ot The motion planning problem is commonly tackled in the space of these para metric representations of rob ot placements or conguration space for short As we will see the conguration space formulation transforms the motion planning prob lem into the problem of nding a continuous curve within a subspace the free space of the conguration 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 ecient solution of one or more pathnding queries The metho ds can b e classied according
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