The aim of tool path planning is to maximize the efficiency against some given precision criteria. In practice, scallop height should be kept constant to avoid unnecessary cutting, while the tool path should be smooth enough to maintain a high feed rate. However, iso-scallop and smoothness often conflict with each other. Existing methods smooth iso-scallop paths one-by-one, which make the final tool path far from being globally optimal. This paper proposes a new framework for tool path optimization. It views a family of iso-level curves of a scalar function defined over the surface as tool path so that desired tool path can be generated by finding the function that minimizes certain energy functional and different objectives can be considered simultaneously. We use the framework to plan globally optimal tool path with respect to iso-scallop and smoothness. The energy functionals for planning iso-scallop, smoothness, and optimal tool path are respectively derived, and the path topology is studied too. Experimental results are given to show effectiveness of the proposed methods.
There also exist some efforts to generate smooth tool paths without considering the overlapping between neighbor machining strips (i.e., the iso-scallop condition). Generally, such methods are based on the Laplacian. For example, Bieterman and Sandstrom @cite_10 proposed a Laplacian based contour parallel tool path generation method by selecting the level sets of a harmonic function defined over a pocket as the tool path. But how to choose the level sets for it still remains an open problem, namely there is no formula for path interval calculation so far. Similarly, Chuang and Yang @cite_34 combined the Laplacian method and iso-parametric method to generate tool paths for pockets with complex topology, i.e., complex boundaries and islands. However, the smoothness of the tool path cannot be guaranteed through Laplacian energy as small Laplacian value does not necessarily mean small curvature of the level set curves. And solving a Laplace equation over a surface can only generate a unique and uncontrollable scalar function (scaling has no impact on the shape of tool paths). Another drawback of the Laplacian based approach is the severe overlapping between machining strips of neighbor paths, especially for paths near the boundary, which results in too much redundant machining.
@cite_34 @cite_10
1998951994 2010512922
A novel curvilinear tool-path generation method is described for planar milling of pockets. The method uses the solution of an elliptic partial differential equation boundary value problem defined on a pocket region. This mathematical function helps morph a smooth low-curvature spiral path in a pocket interior to one that conforms to the pocket boundary. This morphing leads to substantial reductions of tool wear in cutting hard met als and of machining time in cutting all met als, as experiments described here show. A variable feed-rate optimization procedure is also described. This procedure incorporates path, tool-engagement, and machine constraints and can be applied to maximize machine performance for any tool path. This paper presents a novel approach for generating efficient tool paths in machining free-form surfaces. Concept of iso-phote is used to facilitate tool-path generation. An iso-phote is defined as a region on a surface where the normal vector does not differ by more than a prescribed angle from a fixed reference vector. The boundary curves of the iso-phote, called iso-inclination curves, are numerically generated and are served as the initial master tool paths. These iso-inclination curves are then projected to a 2D plane which is perpendicular to the fried reference vector. 2D curve offsetting of the projected iso-inclination curve is then performed. The resulted 2D offset curves are projected back to 3D surface to form final tool paths. The resulted tool paths can guarantee the satisfaction of machining tolerance requirements. A comparison study of this iso-phote based machining with the conventional iso-parametric machining and the iso-planar machining shows favorite result for the new approach. Numerically controlled milling is the primary method for generating complex die surfaces. These complex surfaces are generated by a milling cutter which removes material as it traces out pre-specified tool paths. The accuracy of tool paths directly affects the accuracy of the manufactured surface. The geometry and spacing of the tool paths impact the scallop height and time of manufacturing respectively. In this paper we propose a new method for generating NC tool paths. This method gives the part programmer direct control over the scallop height of the manufactured surface. The method also provides options to the part programmer for generating a variety of tool paths based on practical metrics such as tool path length, tool path curvature and number of tool retractions. This paper presents an analytical method for planning an efficient tool-path in machining free-form surfaces on 3-axis milling machines. This new approach uses a nonconstant offset of the previous tool-path, which guarantees the cutter moving in an unmachined area of the part surface and without redundant machining. The method comprises three steps : (1) the calculation of the tool-path interval, (2) the conversion from the path interval to the parametric interval, and (3) the synthesis of efficient tool-path planning. Presented in the paper is a new approach to tool-path generation for sculptured surface machining. In the proposed C-space approach, the geometric data describing the design-surface, stock-surface and tool shape are transformed into C-space elements, and then, all the tool-path generation decisions are made in the configuration space (C-space). The C-space approach provides a number of distinctive features suitable for sculptured surface machining, including: 1) gouge-free tool-paths; 2) uncut handling; 3) balanced cutting-load; 4) smooth cutter movement; 5) collision-free tool-path. Let M be a planar embedded graph whose arcs meet transversally at the vertices. Let ?(ɛ) be a strip-shaped domain around M, of width ɛ except in a neighborhood of the singular points. Assume that the boundary of ?(ɛ) is smooth. We define comparison operators between functions on ?(ɛ) and on M, and we derive energy estimates for the compared functions. We define a Laplace operator on M which is in a certain sense the limit of the Laplace operator on ?(ɛ) with Neumann boundary conditions. In particular, we show that the p-th eigenvalue of the Laplacian on ?(ɛ) converges to the p-th eigenvalue of the Laplacian on M as ɛ tends to 0. A similar result holds for the magnetic Schrodinger operator. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e., a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling. In this paper, a method for generating boundary-conformed pocketing toolpaths is developed. Based on the 2D Laplace parameterization of pocket contours and the redistribution of the original Laplace isoparametrics, continuous toolpaths are generated. These generated toolpaths have neither thin walls nor leftover tool marks. Detailed algorithms are formulated in steps. The method can be applied to general pockets either with or without islands. Some examples are provided to demonstrate the applicability of this method. In most cases, the method can successfully generate satisfactory toolpaths for arbitrary shaped pockets. However, according to the shape of the pockets and the distribution of the islands, when using this method, over machining may occur in some narrow or bottlenecked areas. Further investigation on how to alleviate this problem is needed. We believe that this method provides an alternative choice for pocket machining. An algorithm for three-axis NC tool path generation on sculptured surfaces is presented. Non-constant parameter tool contact curves are defined on the part by intersecting parallel planes with the part model surface. Four essential elements of this algorithm are introduced: initial chordal approximation, true machining error calculation, direct gouge elimination, and non-constant parameter tool pass interval adjustment. A software implementation of this algorithm produces graphical output depicting the tool path superimposed over the part surface, and it outputs cutter location (CL) data for further post-processing. Several applications examples are presented to demonstrate the capabilities of the algorithm. The results of this technique are compared to those generated from a commercially available computer-aided manufacturing program, and indicate that equivalent accuracy is obtained with many fewer CL points. A novel approach for the NC tool-path generation of free-form surfaces is presented. Traditionally, the distance between adjacent tool-paths in either the Euclidean space or in the parametric space is kept constant. Instead, in this work, the scallop-height is kept constant. This leads to a significant reduction in the size of the CL (cutter location) data accompanied by a reduction in the machining time. This work focuses on the zig-zag (meander) finishing using a ball-end milling cutter. In the last years, the Flight Mechanics Research Group of Politecnico di Torino started a wide research activity, focused on exploration and implementation of path planning algorithms for commercial autopilots, typically adopted on unmanned vehicles. Different path planning approaches were implemented in a Matlab Simulink based tool, generating waypoints sequences with four methods: geometric predefined trajectories, manual waypoints definition, automatic waypoints distribution (i.e. optimizing camera payload capabilities) and, finally, a comprehensive A*-based approach. The tool was also integrated with functions managing the maps used for planning. In this paper, two approaches to path planning in presence of orographic obstacles are detailed. The first algorithm is subdivided in three phases: the generation of a risk map associated with the ground orography, the transformation of the map in a digraph analyzed with the A* algorithm (to obtain the path with minimum risk minimum distance) and finally the smoothing phase, to obtain a flyable waypoint sequence, realized with the Dubins curves. The structure of this method was defined and implemented, but its optimization is still in progress. The second algorithm is based on the same risk map, but optimizing polynomial curves with a genetic algorithm. This method produces a flyable waypoints sequence, minimizing a cost function reflecting path length and collision risk. An extension of this path solver, including aircraft performance constraints, was also considered. This method was tested on sample domains and its computational cost has still to be evaluated before the implementation in the tool. In free-form surface milling, cusps on a part surface need to be regulated. They should be small enough for precision purposes. On the other hand, we should maintain high enough cusps so as not to waste effort making unnecessary cuts. A widely accepted practice is to maintain a constant cusp height over the surface. This paper introduces a new approach to generating constant cusp height tool paths. First, we define a Riemannian manifold by assigning a new metric to a part surface without embedding. This new metric is constructed from the curvature tensors of a part and a tool surface, which we refer to as a cusp-metric. Then, we construct geodesic parallels on the new Riemannian manifold. We prove that a selection from such a family of geodesic parallels constitutes a ''rational'' approximation of accurate constant cusp height tool paths. Verifying critical numerical software involves the generation of test data for floating-point intensive programs. As the symbolic execution of floating-point computations presents significant difficulties, existing approaches usually resort to random or search-based test data generation. However, without symbolic reasoning, it is almost impossible to generate test inputs that execute many paths with floating-point computations. Moreover, constraint solvers over the reals or the rationals do not handle the rounding errors. In this paper, we present a new version of FPSE, a symbolic evaluator for C program paths, that specifically addresses this problem. The tool solves path conditions containing floating-point computations by using correct and precise projection functions. This version of the tool exploits an essential filtering property based on the representation of floating-point numbers that makes it suitable to generate path-oriented test inputs for complex paths characterized by floating-point intensive computations. The paper reviews the key implementation choices in FPSE and the labeling search heuristics we selected to maximize the benefits of enhanced filtering. Our experimental results show that FPSE can generate correct test inputs for selected paths containing several hundreds of iterations and thousands of executable floating-point statements on a standard machine: this is currently outside the scope of any other symbolic-execution test data generator tool. SUMMARY In contrast to most triangulation algorithms which implicitly assume that triangulation point locations are fixed, 'Laplacian' smoothing focuses on moving point locations to improve triangulation. Laplacian smoothing is attractive for its simplicity but it does require an existing triangulation. In this paper the effect of Laplacian smoothing on Delaunay triangulations is explored. It will become clear that constraining Laplacian smoothing to maintain a Delaunay triangulation measurably improves Laplacian smoothing. An early reference to the use of Laplacian smoothing is to be found in a papel by Buell and Bush.' They surveyed the use of Laplace's equation to generate two-dimensional meshes, and described an equipotential method2 which extends finite differences of Laplace's equation on rectangular grids to differences on triangular grids. On triangular grids, the weighted difference equations From a random point O in an infinite strip of width 1, we move in a randomly chosen direction along a curve Г. What shape of Г gives the minimum value to the expectation of the length of the path that reaches the boundary of the strip? After certain arguments suggesting that the desired curve belongs to one of four classes, it is proved that the best curve in those classes consists of four parts: an interval OA of length a, its smooth continuation, an arc AB of radius 1 and small length φ, an interval BD that is smooth continuation of the arc, and an interval DF (with a corner at the point D). If O is the origin and OA is the x axis of a coordinate system, then the coordinates of the above-mentioned points are as follows: A(a, 0), B(a + sin φ, 1 − cos φ), F(a, 1), and @math For the best curve, we have: a ≈ 0.814 and φ ≈ 0.032. Related questions are discussed. Bibliography: 9 titles. The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a quantitative approach to identify their practical differences. In this work, we provide a quantitative assessment of new numerical methods as well as available state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications. We study the family of p-resistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p =1 the p-resistance coincides with the shortest path distance, for p = 2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase transition takes place. There exist two critical thresholds p* and p** such that if p p**, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p* = 1 + 1 (d − 1) and p** = 1 + 1 (d − 2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p* and p** is an artifact of our proofs). We also relate our findings to Laplacian regularization and suggest to use q-Laplacians as regularizers, where q satisfies 1 p* + 1 q =1. We study the problem of finding shortest tours paths for “lawn mowing” and “milling” problems: Given a region in the plane, and given the shape of a “cutter” (typically, a circle or a square), find a shortest tour path for the cutter such that every point within the region is covered by the cutter at some position along the tour path. In the milling version of the problem, the cutter is constrained to stay within the region. The milling problem arises naturally in the area of automatic tool path generation for NC pocket machining. The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general. We give efficient constant-factor approximation algorithms for both problems. In particular, we give a (3+e)-approximation algorithm for the lawn mowing problem and a 2.5-approximation algorithm for the milling problem. Furthermore, we give a simple 65-approximation algorithm for the TSP problem in simple grid graphs, which leads to an 115-approximation algorithm for milling simple rectilinear polygons. Inferring AS-level end-to-end paths can be a valu- able tool for both network operators and researchers. A widely known technique for inferring end-to-end paths is to perform traceroute from sources to destinations. Unfortunately, traceroute requires the access to source machines and is resource consuming. In this paper, we propose two algorithms for AS-level end-to-end path inference. The key idea of our algorithm is to exploit the AS paths appeared in BGP routing tables and infer AS paths based on the ones. In addition, our algorithms infer AS paths on the granularity of destination prefix instead of destination AS. That is, we infer AS paths from any source AS to any destination prefix. This is essential since routing in the Internet is determined based on destination prefixes instead of destination ASs. The validation results show that our algorithm yields accuracy up to 95 for exact match and accuracy up to 97 for path length match. We further extend our algorithm to infer a set of potential AS paths between a source AS and a destination prefix. We find that on average, 86 of inferred AS path sets are accurate in the sense that one of the paths in the set matches the actual AS path. Note that our algorithms require BGP routing tables only and do not require additional data trace or access to either sources or destinations. In addition, we demonstrate that the accuracy of this BGP-based inference approach cannot go beyond 90 . Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues of symmetrical graphs Eigenvalues of subgraphs with boundary conditions Harnack inequalities Heat kernels Sobolev inequalities Advanced techniques for random walks on graphs Bibliography Index.