This paper presents the first randomized approach to kinodynamic planning (also known as trajectory planning or trajectory design). The task is to determine control inputs to drive a robot from an ...

Randomized kinodynamic planning

From the Publisher:
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and
high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Lectures on Discrete Geometry

Computational geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

There are many problems in computational geometry for which the best know algorithms take time @Q(n^2) (or more) in the worst case while only very low lower bounds are known. In this paper we describe a large class of problems for which we prove that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0. We call such problems 3sum-hard. The best known algorithm for the base problem takes @Q(n^2) time. The class of 3sum-hard problems includes problems like: Given a set of lines in the plane, are there three that meet in a point?; or: Given a set of triangles in the plane, does their union have a hole? Also certain visibility and motion planning problems are shown to be in the class. Although this does not prove a lower bound for these problems, there is no hope of obtaining o(n^2) solutions for them unless we can improve the solution for the base problem.

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On a class of O(n2) problems in computational geometry

From the Publisher:
This volume deals with core problems in robotics, like motion planning, sensor-based planning, manipulation, and assembly planning. It also discusses the application of robotics algorithms in other domains, such as molecular modeling, computer graphics, and image analysis.

Algorithms for Robotic Motion and Manipulation

The authors present the first algorithms for computing all placements of (frictionless) point fingers that put a polygonal part in form closure and all placements of point fingers that achieve second-order immobility of a polygonal part. The algorithms run in O(n{sup 2+{var_epsilon}}+K) and O(n{sup 2}log{sup 2}n+K) time in the case of form closure and second-order immobility, respectively, where n is the number of vertices of the polygon, K is the description size of the resulting set of finger placements, and {var_epsilon} is an arbitrarily small constant. The basic of the algorithm is a translation of the problem into geometric searching problems, which are solved using efficient data structures. The results can be extended to the problem of computing all placements of a line and two points that put a polygonal part in form closure. The resulting algorithm runs in O(n{sup 2}log{sup 2} n+K) time, where K is again the description size of the output.

/pdf/computing-immobilizing-grasps-of-polygonal-parts-2a46e5h6qc.pdf

Computing Immobilizing Grasps of Polygonal Parts

The vibratory bowl feeder is the oldest and still most common approach to the automated feeding (orienting) of industrial parts. In this paper, the authors consider a class of vibratory bowl filters that can be described by removing polygonal sections from the track; this class of filters is referred to as traps. For an n-sided polygonal part and an m-sided polygonal trap, an O(n2m log n) algorithm is given to decide whether the part in a specific orientation will safely move across the trap or will fall through the trap and thus be filtered out. For an n-sided convex polygonal part and m-sided convex polygonal trap, this bound is improved to O((n+m) log n). Furthermore, the authors show how to design various trap shapes, ranging from simple traps to general polygons, which will filter out all but one of the different stable orientations of a given part. Although the runtimes of the design algorithms are exponential in the number of trap parameters, many industrial part feeders use few-parameter traps (ba...

/pdf/trap-design-for-vibratory-bowl-feeders-2nmysbfd4j.pdf

Trap Design for Vibratory Bowl Feeders

A depth order on a set of line segments in 3-space is an order such that line segment $a$ comes before line segment $a'$ in the order when $a$ lies below $a'$ or, in other words, when there is a vertical ray that first intersects $a'$ and then intersects $a$. Efficient algorithms for the computation and verification of depth orders of sets of $n$ line segments in 3-space are presented. The algorithms run in time $O(n^{4/3+\varepsilon})$, for any fixed $\varepsilon > 0$. If all line segments are axis-parallel or, more generally, have only a constant number of different orientations, then the sorting algorithm runs in $O(n\log^3 n)$ time and the verification takes $O(n\log^2 n)$ time. The algorithms can be generalized to handle triangles and other polygons instead of line segments. They are based on a general framework for computing and verifying linear orders extending implicitly defined binary relations.

/pdf/computing-and-verifying-depth-orders-3o1xe01bp8.pdf

Computing and Verifying Depth Orders

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of a polygonal part by a sequence of fences. We show that any polygonal part can be oriented by a sequence of fences placed along a conveyor belt, thereby settling a conjecture by Wiegley et al. (1997), and present the first polynomial-time algorithm to compute the shortest such sequence. The algorithm is easy to implement and runs in time O(n3 logn), where n is the number of vertices of the part.

/pdf/computing-fence-designs-for-orienting-parts-5xclkvr2yh.pdf

Mark H. Overmars

Papers

Algorithms for Robotic Motion and Manipulation

Computing Immobilizing Grasps of Polygonal Parts

Trap Design for Vibratory Bowl Feeders

Computing and Verifying Depth Orders

Computing fence designs for orienting parts