scholarly journals Quadratic Constraints on Rigid-Body Displacements

2010 ◽  
Vol 2 (4) ◽  
Author(s):  
J. M. Selig
Keyword(s):  
Rigid Body ◽  
Clifford Algebra ◽  
Geometric Constraint ◽  
Quadratic Constraints ◽  
General Solutions ◽  
Equal Footing ◽  
Möbius Group ◽  
Products Of Subgroups ◽  
Quadratic Hypersurfaces

In this work, the solution to certain geometric constraint problems are studied. The possible rigid displacements allowed by the constraints are shown to be intersections of the Study quadric of rigid-body displacements with quadratic hypersurfaces. The geometry of these constraint varieties is also studied and is found to be isomorphic to products of subgroups in many cases. This information is used to find extremely simple derivations for general solutions to some problems in kinematics. In particular, the number of assembly configurations for RRPS and RRRS mechanisms are found in this way. In order to treat planes and spheres on an equal footing, the Clifford algebra for the Möbius group is introduced.

Computer-Aided Modelling of Cloth Deformation

1996 ◽  
Author(s):  
Y. F. Zhao ◽  
S. T. Tan ◽  
T. N. Wong ◽  
W. J. Chen
Keyword(s):  
Finite Element Method ◽  
Exact Solution ◽  
Finite Element ◽  
Rigid Body ◽  
Present Method ◽  
Geometric Constraint ◽  
Computationally Efficient ◽  
Large Rotation ◽  
Developable Surfaces ◽  
Computer Aided

Abstract A constrained finite element method for modelling cloth deformation is developed. The bending deformation and the geometric constraint of developable surfaces of the cloth objects are considered. The representation of large rotation and the motion of rigid body are described using the current coordinates with the geometric constraint. The effectiveness of the present method is verified by comparing the thread deformation with the exact solution of catenary. Several examples are given to show that the proposed method converges quickly and is thus computationally efficient.

scholarly journals Variational B-rep model analysis for direct modeling using geometric perturbation

2019 ◽  
Vol 6 (4) ◽  
pp. 606-616
Author(s):  
Qiang Zou ◽  
Hsi-Yung Feng
Keyword(s):  
Rigid Body ◽  
Boundary Representation ◽  
Geometric Constraint ◽  
New Method ◽  
Geometric Representation ◽  
Constraint Systems ◽  
Fundamental Difficulty ◽  
Direct Modeling ◽  
Rigid Body Motions ◽  
Constrained Model

Abstract The very recent CAD paradigm of direct modeling gives rise to the need of processing 3D geometric constraint systems defined on boundary representation (B-rep) models. The major issue of processing such variational B-rep models (in the STEP format) is that free motions of a well-constrained model involve more than just rigid-body motions. The fundamental difficulty lies in having a systematic description of what pattern these free motions follow. This paper proposes a geometric perturbation method to study these free motions. This method is a generalization of the witness method, allowing it to directly deal with variational B-rep models represented by the STEP format. This generalization is essentially achieved by using a direct, geometric representation of the free motions, and then expressing the free motions in terms of composites of several basis motions. To demonstrate the effectiveness of the proposed method, a series of comparisons and case studies are presented. Highlights A new method to analyze geometric constraint systems for direct modeling. A generalization of the witness configuration method. A new method to characterize the constraint states of variational B-rep models.

The Application of Geometric Constraint Programming to the Design of Motion Generating Six-Bar Linkages

2012 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Rigid Body ◽  
Constraint Programming ◽  
Synthesis Method ◽  
Solid Modeling ◽  
Single Step ◽  
Geometric Constraint ◽  
Motion Generation ◽  
Stepwise Manner ◽  
Planar Linkages

This paper looks at the application of Geometric Constraint Programming (GCP) to the synthesis of six-bar planar linkages. GCP is a synthesis method that relies on the built-in geometric capabilities of commercial solid-modeling programs to produce linkage designs while operating in the “sketch” mode for these programs. GCP provides the user with the opportunity to create mechanisms in their entirety at multiple design positions. The complexity of analyzing potential defects (such as circuit or branch defects) within a six-bar mechanism poses significant challenges to the user who might try to design such a mechanism in a single step. The methods presented in this paper apply GCP in a stepwise manner to create six-bar linkages that are less likely to suffer from defects than if they were created in a single step. Stepwise approaches are presented for six-bar mechanisms designed to solve a problem involving rigid-body guidance (motion generation). The linkages considered include the Stephenson I, II, and III chains, as well as the Watt I six-bar. The Watt II six-bar is not included since this mechanism’s application to rigid-body guidance can be handled by GCP methods previously developed for four-bar linkages.

Rigid Body Motion Characteristics and Unified Instantaneous Motion Representation of Points, Lines, and Planes

2004 ◽  
Vol 126 (4) ◽  
pp. 593-601 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Yi Zhang
Keyword(s):  
Rigid Body ◽  
Clifford Algebra ◽  
Algebraic Approach ◽  
Body Motion ◽  
Computation Method ◽  
Displacement Operator ◽  
Characteristic Numbers ◽  
Higher Derivatives ◽  
Linear Elements ◽  
Motion Characteristics

The article presents a unified algebraic approach for the modeling of the instantaneous motions of all linear elements, such as points, lines, and planes, embedded in a rigid body. The paper first addresses the Clifford algebra based displacement operator and its higher derivatives from which the coordinate-independent characteristic numbers with simple geometric meaning are defined. With Clifford algebra, the paper also presents the computation method and examples to demonstrate the process of obtaining the displacement operator and the characteristic numbers. Because of the coordinate independent feature, no tedious coordinate transformation typically found in the conventional instantaneous invariants method is needed.

Application of Instantaneous Invariants to the Analysis and Synthesis of Mechanisms

1977 ◽  
Vol 99 (1) ◽  
pp. 97-103 ◽  
Author(s):  
B. Roth ◽  
A. T. Yang
Keyword(s):  
Problem Solving ◽  
Rigid Body ◽  
Rigid Body Motion ◽  
Geometric Constraint ◽  
Body Motion ◽  
Kinematic Synthesis ◽  
Systematic Procedure ◽  
Synthesis Of Mechanisms ◽  
Analysis And Synthesis ◽  
Kinematic Properties

The objective of this paper is to make instantaneous invariants a more accessible tool for problem solving in the field of kinematics. We present a systematic procedure to determine the instantaneous invariants of a rigid body under geometric constraint, develop a process by which kinematic properties of rigid body motion can be expressed in terms of instantaneous invariants, and apply instantaneous invariants to solve typical kinematic synthesis problems. Four examples are given in detail for illustrative purposes.

scholarly journals Clifford algebra of points, lines and planes

Robotica ◽  
2000 ◽  
Vol 18 (5) ◽  
pp. 545-556 ◽  
Author(s):  
J. M. Selig
Keyword(s):  
Computer Vision ◽  
Computer Graphics ◽  
Rigid Body ◽  
Clifford Algebra ◽  
Stereo Vision ◽  
Vision System ◽  
Linear Elements ◽  
Rigid Body Motions ◽  
Rigid Motions ◽  
Assembly Problem

The Clifford algebra for the group of rigid body motions is described. Linear elements, that is points, lines and planes are identified as homogeneous elements in the algebra. In each case the action of the group of rigid motions on the linear elements is found. The relationships between these linear elements are found in terms of operations in the algebra. That is, incidence relations, the conditions for a point to lie on a line for example are found. Distance relations, like the distance between a point and a plane are found. Also the meet and join of linear elements, for example, the line determined by two planes and the plane defined by a line and a point, are found. Finally three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.

Clifford algebra of three dimensional geometry

Robotica ◽  
2002 ◽  
Vol 20 (6) ◽  
pp. 687-697 ◽  
Author(s):  
Glen Mullineux
Keyword(s):  
Rigid Body ◽  
Clifford Algebra ◽  
Projective Geometry ◽  
Geometric Algebra ◽  
Three Dimensional ◽  
Basis Element ◽  
Rigid Body Motions ◽  
Robotic Applications

The use of Clifford or geometric algebra for dealing with three dimensional geometry is discussed. One issue is the representation of the rigid body motions of rotations and translations as elements within the algebra. The approach used is to work with projective geometry and choose the square of an additional basis element to be large (infinite). This allows Euclidean points to be represented as vectors in the algebra and transforms on these to be handled using bivectors. This paper looks at the use of Clifford algebra for handling the types of transforms required in robotic applications. A number of example applications are given.

scholarly journals Rigid Body Dynamics Using Clifford Algebra

2008 ◽  
Vol 20 (1) ◽  
pp. 141-154 ◽  
Author(s):  
J. M. Selig ◽  
E. Bayro-Corrochano
Keyword(s):  
Rigid Body ◽  
Clifford Algebra ◽  
Rigid Body Dynamics ◽  
Body Dynamics

scholarly journals Research on the Method of Multi-point Pressing Adjustment of Space Mechanism Driven by Measurement

2020 ◽  
Vol 316 ◽  
pp. 01002
Author(s):  
Jinlong Zhao ◽  
Chunzhen Ren ◽  
Zhizhuo Cui ◽  
Fuzhou Du
Keyword(s):  
Mathematical Model ◽  
Rigid Body ◽  
Coordinate System ◽  
Large Scale ◽  
Compensation Method ◽  
Geometric Constraint ◽  
Assembly Process ◽  
Point Pressure ◽  
Rigid Body Transformation

In order to solve the problems of difficult assembly and adjustment and poor operation accessibility in the assembly process of multi-point pressure-type large-scale equipment on spacecraft, this paper presents a method of multi-point compact assembly of space mechanism driven by measurement. The method establishes the geometric constraint mathematical model of the equipment pressing mechanism, proposes the assembly coordinate system based on the mounting hole and the compensation method based on the rigid body transformation, and develops the relevant software according to the method. This method has been verified during the installation of the equipment and has achieved good results, which can be used as a reference for other digital assembly methods of aerospace equipment.

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