Anton Gfrerrer

Geometry and kinematics of the Mecanum wheel

Mecanum wheels are used when omnidirectional movability of a vehicle is desired. That means that the vehicle can move along a prescribed path and at the same time rotate arbitrarily around its center. A Mecanum wheel consists of a set of rolls arranged around the wheel axis. In this paper we describe in detail the geometry of these rolls. We derive simple canonical parameterizations of the roll generating curve and the roll surface itself. These parametric representations reveal the geometry of the roll. With their help we can easily find an approximation of the roll surface by a torus for manufacture purposes. Based on the roll parametrization we study the kinematics of a vehicle featured with Mecanum wheels.

Rational interpolation on a hypersphere

Abstract Let a hypersphere Sd−1 in the d-dimensional Euclidean space Ed and n + 1 points Pi on Sd−1 with corresponding parameter values ti be given. In this paper we show that the problem of finding a rational interpolation curve c ⊂ Sd−1 of algebraic order n1 ⩽ n is linear. We further prove that this problem has either exactly one or none solution. We additionally show that in general there exists a solution curve c if n is even. An algorithm, based on a recursive formula, is given, which enables the user to decide whether a solution curve exists or not. In the first case the algorithm yields a parametrization of this curve. Finally we show the geometric invariance and the parameter invariance of these interpolation curves.

Instrumentation and Analysis-Methods for the Measurement of Profiles using Light Sectioning

This paper presents the instrumentation and analysis methods for the measurement of profiles. The object is rotated in front of a light-sectioning system, which measures the local geometry. A least mean square approximation of the local geometry by a circle is performed, the radius of the circle is used as an estimate for the radius of curvature. The vertical position of the upper tangent to the circle is used to estimate the position of the object. It is shown that the period of revolution can be determined from the aperiodic auto-correlation sequence of the eccentricity of the material. The curve corresponding to the profile of the material is a smooth Jordan curve: this fact is used to take advantage of harmonic filtering to suppress noise and perturbations; and to reconstruct the curve from the radius of curvature. The analysis is performed for both constant rate of revolution and constant arc-length; furthermore, the implementation in MAT-LAB is presented. All methods are demonstrated with real data from industrial measurements

A Unified Approach to Direct Kinematics of Some Reduced Motion Parallel Manipulators

After discussing the Study point transformation operator, a unified way to formulate kinematic problems, using “points moving on planes or spheres” constraint equations, is introduced. Application to the direct kinematics problem solution of a number of different parallel Schönflies motion robots is then developed. Certain not widely used but useful tools of algebraic geometry are explained and applied for this purpose. These constraints and tools are also applied to some special parallel robots called ”double triangular” to show that the approach is flexible and universally pertinent to manipulator kinematics in reducing the complexity of some previously achieved solutions. Finally a novel two legged Schönflies architecture is revealed to emphasize that good design is not only essential to good performance but also to easily solved kinematic models. In this example architecture, with double basally actuated legs so as to minimize moving mass, the univariate polynomial solution turns out to be simplest, i.e, of degree two.

Knowledge-Based Design

The chapter “Knowledge-Based Design”, covers the use of template models, integrated calculation and simulation procedures, and automated routines to support product design. Knowledge-based design enables the collection, storage and re-use of expert knowledge, as well as the subsequent integration of know-how into development processes. Using examples from component and assembly development, the chapter elaborates on the potential of enhanced parametric-associative design and knowledge-based engineering used in combination with simultaneously linked calculation procedures.

Car side window kinematics

It is a demanding and time-consuming task to construct the side window mechanism of a car such that the deflections along the seals are minimized. The reason for this is that the window surface S delivered by the stylist, will in general not be movable in a proper way: It is required that the surface tightly moves along the seals. Deflections will necessarily occur no matter how much effort the engineer invests. This article describes how to find a spatial motion that minimizes seal deflection for some given automobile side window surface. Necessary but acceptable modification of the stylists surface S is also addressed, as well as the lifting mechanism design. It is argued that our systematic approach offers considerable savings compared to the prevailing trial-and-error approach.

Study’s Kinematic Mapping — A Tool for Motion Design

Via Study’s kinematic mapping G the 6-parametric Lie group SE(3) of direct Euclidean displacements can be identified with a certain hyperquadric.M 6 in 7-dimensional real projective space. The mapping has nice geometric properties; for instance one parametric rotation groups are represented by straight lines on M 6, coordinate transformations in Euclidean 3-space are represented by special automorphisms of M 6. With the help of G Euclidean kinematics can be considered as a point-geometry in the sense of Felix Klein’s Erlangen program.

Modeling Techniques in CAD

The chapter “Modeling Techniques in CAD”, includes a detailed introduction of design methods within the CAD environment. Structures and strategies of wireframe, surface and solid modeling are presented and discussed in terms of their application in collaborative product development processes. Beyond the application of primary CAD functionalities, this chapter uses specific examples from the automotive industry to present a variety of methods for the efficient creation of mechanical components and assemblies.

Overview of Virtual Product Development

The chapter “Overview of Virtual Product Development”, first provides a summary of the various stages in the life cycle of mechanical products. The main terms, definitions, and methods of computer-aided product development are then introduced. This includes the historical development of computer-aided design, simulation, and data management. In addition, some selected, representative development workflows in automotive engineering are presented and discussed, and the chapter then closes with a brief introduction to the concepts of collaborative product development.

3R Wrist Positioning - a Classical Problem and its Geometric Background

The wrist centre positioning problem (WP) peculiar to the motion produced by the first three joints of a general six revolute jointed (6R), wrist partitioned serial robot and the underlying geometry is reexamined. Conventionally a sequence of six rotational operations, alternately in terms of known geometric parameters and unknown joint angles, expresses the desired position. However the solution can be represented by four intersection points between a fourth order cyclid, and a circle. Properties of the curves of intersection of the cyclid with the absolute plane reveal why the univariate polynomial (UVP) is of fourth degree rather than eighth as indicated by the Bezout number. Simple cyclid geometry makes it convenient to investigate specific 3R positioning architectures and expose degenerate cases.