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.
Last decade has seen a great deal of literature on tool path planning for free-form surfaces, such as iso-parametric method @cite_25 @cite_12 @cite_1 , iso-planar method @cite_19 @cite_9 @cite_18 , iso-scallop method @cite_23 @cite_32 @cite_13 @cite_17 @cite_3 @cite_8 @cite_21 , iso-phote method @cite_4 and C-space method @cite_33 , to name a few. Surveys of much more work about tool path planning research can be found in @cite_0 @cite_30 . Since we aim at optimal tool paths with respect to iso-scallop and smoothness, we put special interest in the iso-scallop method, which means the height of the points at the scallop curves remains as high as a given value so that the tool path has no unnecessary cutting. Conventionally, constant scallop height is obtained by varying the offset magnitude along each path. A mathematical method for generating iso-scallop tool paths following such strategy was first proposed by @cite_23 . Afterwards, methods to improve the computing efficiency @cite_2 @cite_3 and accuracy @cite_16 @cite_17 @cite_21 were proposed. In 2007, Kim @cite_27 reformulated the iso-scallop tool path as geodesic parallel curves on the design surface by defining a new Riemannian metric.
@cite_30 @cite_3 @cite_2 @cite_18 @cite_4 @cite_8 @cite_21 @cite_23 @cite_17 @cite_32 @cite_19 @cite_27 @cite_16 @cite_25 @cite_12 @cite_33 @cite_9 @cite_1 @cite_0 @cite_13
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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. 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. 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. 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. In this paper, we propose a metric rectification method to restore an image from a single camera-captured document image. The core idea is to construct an isometric image mesh by exploiting the geometry of page surface and camera. Our method uses a general cylindrical surface (GCS) to model the curved page shape. Under a few proper assumptions, the printed horizontal text lines are shown to be line convergent symmetric. This property is then used to constrain the estimation of various model parameters under perspective projection. We also introduce a paraperspective projection to approximate the nonlinear perspective projection. A set of close-form formulas is thus derived for the estimate of GCS directrix and document aspect ratio. Our method provides a straightforward framework for image metric rectification. It is insensitive to camera positions, viewing angles, and the shapes of document pages. To evaluate the proposed method, we implemented comprehensive experiments on both synthetic and real-captured images. The results demonstrate the efficiency of our method. We also carried out a comparative experiment on the public CBDAR2007 data set. The experimental results show that our method outperforms the state-of-the-art methods in terms of OCR accuracy and rectification errors. The continuation of a trend towards more user-friendly, interactive CADCAM systems, prompted the creation of a prototype software package called CISPA (Computer Interactive Surfaces Pre-APT). This system uses a menu-driven front end with graphical feedback to guide a user through curve and free form surface definition resulting in a mathematical model which may be used to generate NC machine CLSF. This front end was built upon the curve and surface complex taken from CAM-I's CASPA (Computer-Aided Sculptured Pre-APT). This complex is part of the execution phase of APT 4. Improvements were made in the definition and calculation of step size basing the input on geometric attributes such as chordal deviation and scallop height rather than on abstract algebraic quantities. Further additions include increased flexibility in milling surfaces, by roughing to depth. 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. 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. 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. We present a new interactive method for tubular structure extraction. The main application and motivation for this work is vessel tracking in 2D and 3D images. The basic tools are minimal paths solved using the fast marching algorithm. This allows interactive tools for the physician by clicking on a small number of points in order to obtain a minimal path between two points or a set of paths in the case of a tree structure. Our method is based on a variant of the minimal path method that models the vessel as a centerline and surface. This is done by adding one dimension for the local radius around the centerline. The crucial step of our method is the definition of the local metrics to minimize. We have chosen to exploit the tubular structure of the vessels one wants to extract to built an anisotropic metric. The designed metric is well oriented along the direction of the vessel, admits higher velocity on the centerline, and provides a good estimate of the vessel radius. Based on the optimally oriented flux this measure is required to be robust against the disturbance introduced by noise or adjacent structures with intensity similar to the target vessel. We obtain promising results on noisy synthetic and real 2D and 3D images and we present a clinical validation. Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of “curviness”: we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents. High-quality volumetric parameterization of computational domain plays an important role in three-dimensional isogeometric analysis. Reparameterization technique can improve the distribution of isoparametric curves surfaces without changing the geometry. In this paper, using the reparameterization method, we investigate the high-quality construction of analysis-suitable NURBS volumetric parameterization. Firstly, we introduce the concept of volumetric reparameterization, and propose an optimal Mobius transformation to improve the quality of the isoparametric structure based on a new uniformity metric. Secondly, from given boundary NURBS surfaces, we present a two-stage scheme to construct the analysis-suitable volumetric parameterization: in the first step, uniformity-improved reparameterization is performed on the boundary surfaces to achieve high-quality isoparametric structure without changing the shape; in the second step, from a new variational harmonic metric and the reparameterized boundary surfaces, we construct the optimal inner control points and weights to achieve an analysis-suitable NURBS solid. Several examples with complicated geometry are presented to illustrate the effectiveness of proposed methods. Motivated by drawing route sketches, we consider the d-regular path schematization problem. For this problem we are given an embedded path P (e.g., a route in a road network) and a positive integer d. The goal is to find a d-schematized embedding of P in which the orthogonal order of all vertices in the input is preserved and in which every edge has a direction that is an integer multiple of (90 d)^o. We show that deciding whether a path can be d-schematized is NP-complete for any positive integer d. Despite the NP-hardness of the problem we still want to be able to generate route sketches and thus need to solve the d-regular path schematization problem. We explore two different algorithmic approaches, both of which consider two additional quality constraints: We require that every edge is drawn with a user-specified minimum length and we want to maximize the number of edges that are drawn with their preferred direction. The first algorithmic approach restricts the input paths to be axis-monotone. We show that there exists a polynomial-time algorithm that solves the d-regular path schematization problem for this restricted class of embedded paths. We extend this approach by a heuristic such that it can handle arbitrary simple paths. However, for the second step we cannot guarantee that the orthogonal order of the input embedding is maintained. The second approach is a formulation of the d-regular path schematization problem as a mixed integer linear program. Finally, we give an experimental evaluation which shows that both approaches generate reasonable route sketches for real-world data. We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the first provably intractable problem in robotics. There is extensive literature on using convex optimization to derive piece-wise polynomial trajectories for controlling differential flat systems with applications to three-dimensional flight for Micro Aerial Vehicles. In this work, we propose a method to formulate trajectory generation as a quadratic program (QP) using the concept of a Safe Flight Corridor (SFC). The SFC is a collection of convex overlapping polyhedra that models free space and provides a connected path from the robot to the goal position. We derive an efficient convex decomposition method that builds the SFC from a piece-wise linear skeleton obtained using a fast graph search technique. The SFC provides a set of linear inequality constraints in the QP allowing real-time motion planning. Because the range and field of view of the robot's sensors are limited, we develop a framework of Receding Horizon Planning , which plans trajectories within a finite footprint in the local map, continuously updating the trajectory through a re-planning process. The re-planning process takes between 50 to 300 ms for a large and cluttered map. We show the feasibility of our approach, its completeness and performance, with applications to high-speed flight in both simulated and physical experiments using quadrotors. 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. Few computational models have addressed the spatiotemporal features of unconstrained three-dimensional (3D) arm motion. Empirical observations made on hand paths, speed profiles, and arm postures during point-to-point movements led to the assumption that hand path and arm posture are independent of movement speed, suggesting that the geometric and temporal properties of movements are decoupled. In this study, we present a computational model of 3D movements for an arm with four degrees of freedom based on the assumption that optimization principles are separately applied at the geometric and temporal levels of control. Geometric properties (path and posture) are defined in terms of geodesic paths with respect to the kinetic energy metric in the Riemannian configuration space. Accordingly, a geodesic path can be generated with less muscular effort than on any other, nongeodesic path, because the sum of all configuration-speed-dependent torques vanishes. The temporal properties of the movement (speed) are determined in task space by minimizing the squared jerk along the selected end-effector path. The integration of both planning levels into a single spatiotemporal representation simplifies the control of arm dynamics along geodesic paths and results in movements with near minimal torque change and minimal peak value of kinetic energy. Thus, the application of Riemannian geometry allows for a reconciliation of computational models previously proposed for the description of arm movements. We suggest that geodesics are an emergent property of the motor system through the exploration of dynamical space. Our data validated the predictions for joint trajectories, hand paths, final postures, speed profiles, and driving torques. In this paper, we propose a generic framework for 3D surface remeshing. Based on a metric-driven Discrete Voronoi Diagram construction, our output is an optimized 3D triangular mesh with a user-defined vertex budget. Our approach can deal with a wide range of applications, from high-quality mesh generation to shape approximation. By using appropriate metric constraints, the method generates isotropic or anisotropic elements. Based on point sampling, our algorithm combines the robustness and theoretical strength of Delaunay criteria with the efficiency of an entirely discrete geometry processing. Besides the general described framework, we show the experimental results using isotropic, quadric-enhanced isotropic, and anisotropic metrics, which prove the efficiency of our method on large meshes at a low computational cost.