Tony C. Woo

Parting directions for mould and die design

On the basis, of the condition for demouldability, two levels of visibility, complete and partial visibility, are defined. The viewing directions from which a surface is completely visible can be represented as a convex region on the unit sphere called the visibility map of the surface. Algorithms are given for dividing a given object into pockets, for which visibility and demouldability can be determined independently, for constructing visibility maps, and for selecting an optimal pair of parting directions for a mould that minimizes the number of cores. An example illustrates the algorithms.

Visibility maps and spherical algorithms

Abstract By the extraction of ideas from computer vision, geometrical design and complexity analysis, a structure called visibility emerges. The paper describes a way in which a 3D workpiece is mapped onto the unit sphere, and its visibility is determined. For applications, manufacturing machines are classified by their degrees of freedom into point, line and surface visible processes Algorithms for optimal workpiece orientation are then formulated as simple intersections on the sphere.

Review of dimensioning and tolerancing: representation and processing

The paper surveys the current state of knowledge of techniques for representing, manipulating and analysing dimensioning and tolerancing data in computer-aided design and manufacturing. The use of solid models and variational geometry, and its implications for the successful integration of CAD and CAM, are discussed. The topics explored so far can be grouped into four categories: (a) the representation ot dimensioning and tolerancing (D& T), (b) the synthesis and analysis of D& T, (() tolerance control, and (d) the implications of D& T in CAM. The paper describes in detail the recent work in each group, and concludes with speculation on a general framework k)r future research.

A Combinatorial Analysis of Boundary Data Structure Schemata

Speed up the software in geometric algorithms for solid modeling, CAD/CAM, and robotics applications. How? By using boundary data structures that are fast and use less storage.

Optimal algorithms for symmetry detection in two and three dimensions

Exact algorithms for detecting all rotational and involutional symmetries in point sets, polygons and polyhedra are described. The time complexities of the algorithms are shown to be θ (n) for polygons and θ (n logn) for two- and three-dimensional point sets. θ (n logn) time is also required for general polyhedra, but for polyhedra with connected, planar surface graphs θ (n) time can be achieved. All algorithms are optimal in time complexity, within constants.

Dimensional measurement of surfaces and their sampling

The number of the discrete samples for the dimensional measurement of machined surfaces and their coordinates is investigated. Counter to intuition, there need not be quadratically more samples than in the case for sampling lines or curves. To justify this novel scheme, accuracy is defined as the discrepancy of a finite point set. Then, from number theory, a particular sequence of numbers is used to compute the sampling coordinates, resulting in a number that is linear in 1D, at the same level of accuracy that is achieved by a 2D uniform distribution. Finally, experimental results of the measurement of machined surfaces modeled as random processes are compiled. coordinate measurement, optical scanning, surface roughness, lowdiscrepancy point sets

Separating and intersecting spherical polygons: computing machinability on three-, four-, and five-axis numerically controlled machines

We consider the computation of an optimal workpiece orientation allowing the maximal number of surfaces to be machined in a single setup on a three-, four-, or five-axis numerically controlled machine. Assuming the use of a ball-end cutter, we establish the conditions under which a surface is machinable by the cutter aligned in a certain direction, without the cutters being obstructed by portions of the same surface. The set of such directions is represented on the sphere as a convex region, called the visibility map of the surface. By using the Gaussian maps and the visibility maps of the surfaces on a component, we can formulate the optimal workpiece orientation problems as geometric problems on the sphere. These and related geometric problems include finding a densest hemisphere that contains the largest subset of a given set of spherical polygons, determining a great circle that separates a given set of spherical polygons, computing a great circle that bisects a given set of spherical polygons, and finding a great circle that intersects the largest or the smallest subset of a set of spherical polygons. We show how all possible ways of intersecting a set of n spherical polygons with v total number of vertices by a great circle can be computed in O(vn log n) time and represented as a spherical partition. By making use of this representation, we present efficient algorithms for solving the five geometric problems on the sphere.

Spherical maps: their construction, properties, and approximation

The Gaussian map and its allied visibility map on a unit sphere find wide applications for orientating the workpiece for machining by numerical control machines and for probing by coordinate measurement machines. They also provide useful aids in computerized scene analysis, computation of surface-surface intersection, component design for manufacturing and fabrication procedures. Spherical convex hulls and spherical circles are two geometric constructs used to approximate the Gaussian maps and the visibility maps. The duality and the complementarity of these spherical maps are examined so as to derive efficient algorithms.

Sensor Optimization for Fault Diagnosis in Single Fixture Systems: A Methodology

Fixture fault diagnosis is a critical component of currently evolving techniques aimed at manufacturing variation reduction. The impact of sensor location on the effectiveness of fault-type discrimination in such diagnostic procedures is significant. This paper proposes a methodology for achieving optimal fault-type discrimination through an optimized configuration of defined “sensor locales.” The optimization is presented in the context of autobody fixturing—a predominant cause of process variability in automobile assembly. The evaluation criterion for optimization is an improvement in the ability to provide consistency of best match, in a pattern recognition sense, of any fixture error to a classified, anticipated error set. The proposed analytical methodology is novel in addressing optimization by incorporating fixture design specifications in sensor locale planning—constituting a Design for Fault Detectability approach. Examples of the locale planning for a single fixture sensor layout and an application to an industrial fixture configuration are presented to illustrate the proposed methodology.

Real time motion fairing with unit quaternions

Abstract Though it may be tempting to smooth orientation data by filtering the Euler angles directly, it is noted that smoothed Euler angles do not necessarily yield a smooth motion. This is caused by the difference between the metric in the rotation group and that in the Euclidean space. The quaternions, which Hamilton discovered in 1853, provide a means for representing rotation. A unit quaternion, represented as a hypersphere in R 4, has the same local topology and geometry as the rotation group. It thus provides a means for interpolating orientations. It is possible to achieve smooth rotation by filtering in quaternions the resulting quaternion may no longer be unitized. Fortunately, a unit quaternion curve, which represents the rotation path, can be derived by integrating the exponential map of the angular velocity. Unity of quaternions is thus maintained by filtering angular velocities. A lowpass filter coupled with an adaptive, mediative filter are employed to achieve smooth rotation motion in real time.