A Note on the Diagonalizability and the Jordan Form of the 4×4 Homogeneous Transformation Matrix

2005 ◽  
Vol 128 (6) ◽  
pp. 1343-1348
Author(s):  
Sangamesh Deepak R ◽  
Ashitava Ghosal
Keyword(s):  
Geometric Modeling ◽  
Parallel Manipulators ◽  
Transformation Matrix ◽  
Jordan Form ◽  
Screw Axis ◽  
General Transformation ◽  
3D Space ◽  
Homogeneous Transformation Matrix ◽  
Jordan Basis ◽  
Aided Design

The 4×4 homogeneous transformation matrix is extensively used for representing rigid body displacement in 3D space and has been extensively used in the analysis of mechanisms, serial and parallel manipulators, and in the field of geometric modeling and computed aided design. The properties of the transformation matrix are very well known. One of the well known properties is that a general 4×4 homogeneous transformation matrix cannot be diagonalized, and at best can be reduced to a Jordan form. In this paper, we show that the 4×4 homogeneous transformation matrix can be diagonalized if and only if displacement along the screw axis is zero. For the general transformation with nonzero displacement along the axis, we present an explicit expression for the fourth basis vector of the Jordan basis. We also present a variant of the Jordan form which contains the motion variables along and about the screw axis and the corresponding basis vectors which contains the information only about the screw axis and its location. We present a novel expression for a point on the screw axis closest to the origin, which is then used to form a simple choice of basis for different forms. Finally, the theoretical results are illustrated with a numerical example.

scholarly journals Advances in Engineering Graphics: Improvements and New Proposals

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 827
Author(s):  
José Ignacio Rojas-Sola
Keyword(s):  
Computer Animation ◽  
Computer Aided Design ◽  
Geometric Modeling ◽  
Correct Interpretation ◽  
Special Issue ◽  
Engineering Graphics ◽  
Communication Techniques ◽  
Graphic Communication ◽  
Graphic Language ◽  
Aided Design

The study of graphic communication techniques that engineers, architects, and designers use to express ideas and concepts, or the graphic expression applied to the design process, is becoming increasingly important. The correct interpretation of graphic language allows the development of skills in the training of an engineer or architect. For this reason, research on this topic is especially valuable in finding improvements or new proposals that help toward a better understanding of those techniques. This Special Issue shows the reader some examples of different disciplines available, such as engineering graphics, industrial design, geometric modeling, computer-aided design, descriptive geometry, architectural graphics and computer animation.

Determination of the Workspace of a Three-Degrees-of-Freedom Parallel Manipulator Using a Three-Dimensional Computer-Aided-Design Software Package and the Concept of Virtual Chains1

2015 ◽  
Vol 8 (2) ◽  
Author(s):  
Andrew Johnson ◽  
Xianwen Kong ◽  
James Ritchie
Keyword(s):  
Computer Aided Design ◽  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Graphical Representation ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Workspace Analysis ◽  
Computer Aided ◽  
Aided Design

The determination of workspace is an essential step in the development of parallel manipulators. By extending the virtual-chain (VC) approach to the type synthesis of parallel manipulators, this technical brief proposes a VC approach to the workspace analysis of parallel manipulators. This method is first outlined before being illustrated by the production of a three-dimensional (3D) computer-aided-design (CAD) model of a 3-RPS parallel manipulator and evaluating it for the workspace of the manipulator. Here, R, P and S denote revolute, prismatic and spherical joints respectively. The VC represents the motion capability of moving platform of a manipulator and is shown to be very useful in the production of a graphical representation of the workspace. Using this approach, the link interferences and certain transmission indices can be easily taken into consideration in determining the workspace of a parallel manipulator.

A General Approach for Geometric Error Modeling of Lower Mobility Parallel Manipulators

2011 ◽  
Vol 3 (2) ◽  
Author(s):  
Haitao Liu ◽  
Tian Huang ◽  
Derek G. Chetwynd
Keyword(s):  
Systematic Approach ◽  
Parallel Manipulators ◽  
Geometric Error ◽  
Error Modeling ◽  
Kinematic Calibration ◽  
Accuracy Improvement ◽  
Linear Map ◽  
Homogeneous Transformation Matrix ◽  
Lower Mobility ◽  
Source Errors

This paper presents a general and systematic approach for geometric error modeling of lower mobility manipulators. The approach can be implemented in three steps: (1) development of a linear map between the pose error twist and source errors within an individual limb using the homogeneous transformation matrix method; (2) formulation of a linear map between the pose error twist and the joint error intensities of a lower mobility parallel manipulator; and (3) combination of these two models. The merit of this approach lies in that it enables the source errors affecting the compensatable and uncompensatable pose accuracy of the platform to be explicitly separated, thereby providing designers and/or field engineers with an informative guideline for the accuracy improvement achievable by suitable measures, i.e., component tolerancing in design, manufacturing and assembly processes, and kinematic calibration. Three typical and well-known parallel manipulators are taken as examples to illustrate the generality and effectiveness of this approach.

Instantaneous Screws of Weight-Bearing Knee: What Can the Screws Tell Us About the Knee Motion

2014 ◽  
Vol 136 (7) ◽  
Author(s):  
Alon Wolf
Keyword(s):  
Screw Theory ◽  
Weight Bearing ◽  
Theoretical Background ◽  
Geometrical Method ◽  
Screw Axis ◽  
Knee Motion ◽  
3D Space ◽  
Instantaneous Screw Axis ◽  
Instantaneous Screw ◽  
Insight Into

There are several ways to represent a given object's motion in a 3D space having 6DOF i.e., three translations and three rotations. Some of the methods that are used are mathematical and do not provide any geometrical insight into the nature of the motion. Screw theory is a mathematical, while at the same time, geometrical method in which the 6DOF motion of an object can be represented. We describe the 6DOF motion of a weight-bearing knee by its screw parameters, that are extracted from 3D Optical Reflective motion capture data. The screw parameters which describe the transformation of the shank with respect to the thigh in each two successive frames, is represented as the instantaneous screw axis of the motion given in its Plücker line coordinate, along with its corresponding pitch and intensity values. Moreover, the Striction curve associated with the motion provides geometrical insight into the nature of the motion and its repeatability. We describe the theoretical background and demonstrate what the screw can tell us about the motion of healthy subjects' knee.

A Quadratic Trigonometric Nu Spline with Shape Control

2017 ◽  
Vol 17 (03) ◽  
pp. 1750015 ◽  
Author(s):  
Muhammad Sarfraz ◽  
Shamaila Samreen ◽  
Malik Zawwar Hussain
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Shape Control ◽  
Partition Of Unity ◽  
Control Points ◽  
Geometric Continuity ◽  
Spline Method ◽  
Shape Effects ◽  
Curve Modeling ◽  
Aided Design

A significant curve modeling technique has been introduced with a view to its applications in geometric modeling, computer graphics and computer-aided design. It is a new spline method using quadratic trigonometric functions with well-controlled shape influences of parameters introduced through geometric continuity of order two. The proposed curve model owns the best possible geometric properties such as convex hull, partition of unity, affine invariance and variation diminishing. A quadratic normally has three control points giving lesser flexibility to one piece of curve. However, a cubic has four control points giving higher flexibility to one piece of curve. In the proposed scheme, we have introduced a quadratic trigonometric with four control points. Thus, the proposed quadratic trigonometric has embedded geometric features of cubic/cubic trigonometric. The proposed spline method, constrained with Nu spline like GC2 smoothness, produces a quadratic trigonometric Nu spline (QTNS) with interesting shape control locally and globally. The method is helpful for a variety of shape effects, like point tension, interval tension or global tension. It also produces a quadratic trigonometric alternative to cubic/cubic trigonometric spline because of having four control points in its piecewise description.

scholarly journals Usability Studies In Virtual And Traditional Computer Aided Design Environments For Navigation

2013 ◽  
Vol 17 (4) ◽  
pp. 205
Author(s):  
Syed Adeel Ahmed ◽  
Kurt M. Satter
Keyword(s):  
Virtual Environment ◽  
Computer Aided Design ◽  
Tracking System ◽  
User Preferences ◽  
Usability Study ◽  
3D Space ◽  
Human Computer Interfaces ◽  
Spatial Environment ◽  
Navigation Test ◽  
Aided Design

A usability study was used to measure user performance and user preferences for a CAVETM immersive stereoscopic virtual environment with wand interfaces compared directly with a workstation non-stereoscopic traditional CAD interface with keyboard and mouse. In both the CAVETM and the adaptable technology environments, crystal eye glasses are used to produce a stereoscopic view. An ascension flock of birds tracking system is used for tracking the users head and wand pointing device positions in 3D space. It is argued that with these immersive technologies, including the use of gestures and hand movements, a more natural interface in immersive virtual environments is possible. Such an interface allows a more rapid and efficient set of actions to recognize geometry, interaction within a spatial environment, the ability to find errors, and navigate through a virtual environment. The wand interface provides a significantly improved means of interaction. This study quantitatively measures the differences in interaction when compared with traditional human computer interfaces. This paper provides analysis via usability study methods for navigation termed as Benchmark 1. During testing, testers are given some time to play around with the CAVETM environment for familiarity before undertaking a specific exercise. The testers are then instructed regarding tasks to be completed, and are asked to work quickly without sacrificing accuracy. The research team timed each task, and recorded activity on evaluation sheets for Navigation Test. At the completion of the testing scenario involving navigation, the subject/testers were given a survey document and asked to respond by checking boxes to communicate their subjective opinions.

Computer-Aided Modeling and Manufacturing of Spherical Mechanisms via a Novel Web Tool

2007 ◽  
Vol 7 (4) ◽  
pp. 339-346 ◽  
Author(s):  
J. Schuler ◽  
J. Ketchel ◽  
P. Larochelle
Keyword(s):  
Dynamic Simulation ◽  
Computer Aided Design ◽  
Geometric Modeling ◽  
Three Dimensional ◽  
Software Tool ◽  
Web Based ◽  
Spherical Mechanisms ◽  
Computer Aided ◽  
Manufacturing Software ◽  
Aided Design

In this paper, we present a novel web-based computer-aided modeling and manufacturing software tool for spherical mechanisms. Our purpose is to facilitate the analysis, dynamic simulation, and manufacture of one degree of freedom spherical four-bar mechanisms. First, a brief review of some of the current computer-aided design software for spherical four-bar mechanisms is presented. These software packages provide the three-dimensional visualization and computational capabilities necessary to synthesize and analyze spherical four-bar mechanisms. However, to date, no readily available and effective tools exist to aid in the modeling and manufacture of spherical mechanisms. Next, the kinematics of spherical four-bar mechanisms are reviewed as they pertain to their geometric modeling and manufacture. Finally, we present our web-based implementation of a computer-aided modeling, simulation, and manufacturing methodology for spherical four-bar mechanisms called SFBDESIGNER (for spherical four-bar designer). SFBDESIGNER facilitates the design, dynamic simulation, prototyping, and manufacture of spherical four-bar mechanisms.

Solving reachable workspace of some parallel manipulators by computer-aided design variation geometry

2008 ◽  
Vol 222 (9) ◽  
pp. 1773-1781 ◽  
Author(s):  
Y Lu ◽  
Y Shi ◽  
B Hu
Keyword(s):  
Computer Aided Design ◽  
Degrees Of Freedom ◽  
Parallel Manipulators ◽  
Six Degrees Of Freedom ◽  
Reachable Workspace ◽  
Spline Curves ◽  
Computer Aided ◽  
Design Variation ◽  
Aided Design ◽  
The Given

To shape the workspace of some novel parallel manipulators (PMs) is significant. A novel computer-aided design (CAD) variation geometry approach is proposed to shape and solve the reachable workspace of some PMs with three to six degrees of freedom (DOFs). Some basic techniques are described for designing the simulation mechanism and solving the reachable workspace. The simulation mechanisms of some PMs with three to six DOFs are created. When varying the driving dimensions of the active legs in the given extent, the simulation mechanisms vary correspondingly, and the position components of the moving platform are solved automatically. By transferring the position solutions into spatial spline curves in the simulation mechanism, all the boundary surfaces of the workspace can be created and visualized dynamically. Comparing with analytic approaches for solving workspace, the CAD variation geometry approach is simple, straightforward, accurate, and repeatable.

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