Solution Selectors: A User-Oriented Answer to the Geometric Constraint Multiple Solution Problem

2001 ◽  
Author(s):  
Bernhard Bettig ◽  
Jami Shah
Keyword(s):  
Linear Equations ◽  
Independent Solution ◽  
Solid Modeling ◽  
Multiple Solution ◽  
Parametric Modeling ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Basic Solution ◽  
Solution Selection ◽  
Solution Problem

Abstract The development of solid modeling to represent the geometry of designed parts and the development of parametric modeling to control the size and shape have had significant impacts on the efficiency and speed of the design process. Designers now rely on parametric solid modeling, but surprisingly often are frustrated by a problem that unpredictably causes their sketches to become twisted and contorted. This problem, known as the “multiple solution problem” occurs because the dimensions and geometric constraints yield a set of non-linear equations with many roots. This situation occurs because the dimensioning and geometric constraint information given in a CAD model is not sufficient to unambiguously and flexibly specify which configuration the user desires. This paper first establishes that only explicit, independent solution selection declarations can provide a flexible mechanism that is sufficient for all situations of solution selection. The paper then describes the systematic derivation of a set of “solution selector” types by considering the occurrences of multiple solutions in combinations of mutually constrained geometric entities. The result is a set of eleven basic solution selector types and two derived types that incorporate topological information. In particular, one derived type “concave/convex” is user-oriented and thought to be very useful.

Solution Selectors: A User-Oriented Answer to the Multiple Solution Problem in Constraint Solving

2003 ◽  
Vol 125 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Bernhard Bettig ◽  
Jami Shah
Keyword(s):  
Linear Equations ◽  
Independent Solution ◽  
Solid Modeling ◽  
Multiple Solution ◽  
Parametric Modeling ◽  
Geometric Constraints ◽  
Basic Solution ◽  
Solution Selection ◽  
Systematic Derivation ◽  
Solution Problem

The development of solid modeling to represent the geometry of designed parts and the development of parametric modeling to control the size and shape have had significant impacts on the efficiency and speed of the design process. Designers now rely on parametric solid modeling, but often are frustrated by a problem that unpredictably causes their sketches to become twisted, contorted, or take an unexpected shape. Mathematically, this problem, known as the “multiple solution problem” occurs because the dimensions and geometric constraints yield a set of non-linear equations with many roots. In practice, this situation occurs because the dimensioning and geometric constraint information given in a CAD model is not sufficient to unambiguously and flexibly specify which configuration the user desires. This paper first establishes that only explicit, independent solution selection declarations can provide a flexible mechanism that is sufficient for all situations. The paper then describes the systematic derivation of a set of “solution selector” types by considering the occurrences of multiple solutions in combinations of mutually constrained geometric entities. The result is a set of eleven basic solution selector types and two derived types that incorporate topological information. In particular, one derived type “concave/convex” is user-oriented and may prove to be particularly useful.

Parametric Solid Modeling

2006 ◽  
Author(s):  
Horea T. Ilies¸
Keyword(s):  
Heuristic Algorithms ◽  
Solid Modeling ◽  
Parametric Modeling ◽  
Geometric Constraints ◽  
Parametric Models ◽  
Open Problems ◽  
Solid Models ◽  
Modeling Systems ◽  
Alternative Approaches ◽  
And Performance

Parametric modeling systems are fundamentally changing the design process practiced in the industry today. Practically all commercial CAD systems combine established solid modeling techniques with constraint solving and heuristic algorithms to create, edit and manipulate solid models, while enforcing the requirement that every such solid model must maintain the validity of the prescribed geometric constraints. However, a number of fundamental (open) problems limit the functionality and performance of these parametric modeling systems. For example, the allowable parametric changes are history dependent; the number of parameters describing even relatively simple parts can quickly become prohibitively large, and commercial constraint solvers are limited today to 2-dimensional geometric constraints. Consequently, current parametric modeling systems do not support many practical design situations due to the associated theoretical and computational difficulties, as well as to the considerable organizational obstacles generated by the need to handle large parametric models. This paper investigates the current practices and limitations of parametric solid modeling systems, and explores some alternative approaches that could complement the identified limitations.

Function Generation With Finitely-Separated Precision Points Using Geometric Constraint Programming

2006 ◽  
Author(s):  
Edward C. Kinzel ◽  
James P. Schmiedeler ◽  
Gordon R. Pennock
Keyword(s):  
Constraint Programming ◽  
Computer Aided Design ◽  
Linear Equations ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Function Generation ◽  
Design Software ◽  
Aided Design ◽  
Four Bar Linkage ◽  
Non Linear Equations

This paper explains how Geometric Constraint Programming can be applied to solve function generation problems with finitely-separated positions using a number of different mechanisms. Geometric Constraint Programming uses the sketching mode of commercial parametric computer-aided design software to create kinematic diagrams whose elements are parametrically related so that when a parameter is changed, the design is modified automatically. Geometric constraints are imposed graphically through the user interface, and the numerical solvers integrated into the software solve the relevant systems of non-linear equations without the user explicitly formulating those equations. A key advantage of using Geometric Constraint Programming for function generation is that the same approach can be applied to any mechanism, so no unique algorithms are required. Furthermore, because the implementation is relatively straightforward regardless of the chosen mechanism, the designer can quickly and easily generate solutions for a large number of precision points and/or with complex mechanisms to provide a very accurate match to the desired function. Examples of function generation with a four-bar linkage, a six-bar linkage, and a seven-bar linkage illustrate the benefits of the proposed methodology.

INTERACTION WITH CONSTRAINTS IN 3D MODELING

1991 ◽  
Vol 01 (04) ◽  
pp. 405-425 ◽  
Author(s):  
WOLFGANG SOHRT ◽  
BEAT D. BRÜDERLIN
Keyword(s):  
3D Modeling ◽  
Degrees Of Freedom ◽  
Solid Modeling ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Interactive Modeling ◽  
Constraint Solver ◽  
Graphical Interaction ◽  
Definition Of

This paper presents an implementation of an interactive solid modeling system that integrates 1) the definition of objects by graphical interaction and 2) the specification of objects by geometric constraints. In this system, interactive modeling operations for constructing assemblies automatically generate constraints to maintain the properties intended by their invocation, and constraints, in turn, determine the degrees of freedom for further interactive mouse-controlled modeling operations. A symbolic geometric constraint solver is employed for solving systems of simultaneous constraints. Group hierarchies are utilized for representing dependencies and for localizing systems of constraints.

Mixed Exact and Approximate Position Design of Planar Linkages via Geometric Constraint Programming (GCP) Techniques

2015 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Undergraduate Students ◽  
Constraint Programming ◽  
Parametric Modeling ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Target Zones ◽  
Industry Standard ◽  
Modeling Software ◽  
Moving Points ◽  
Circular Target

The synthesis of mechanisms to reach multiple positions can often be satisfied by the specification of a combination of exact and approximate positions. Geometric Constraint Programming (GCP) uses industry standard parametric modeling software to obtain solutions to planar synthesis problems. This paper demonstrates the capability of GCP to solve problems that contain a combination of exact and approximate positions. The approximate positions are added to existing GCP design approaches by the application of geometric constraints to locate moving points on a mechanism within specified circular target zones. The target zones are used to guide the coupler point of a linkage along an approximate path between critical precision positions. The approach applies to the synthesis of both four-bar and complex linkages. In complex linkages, the target zones can be applied to multiple points on the linkage to better coordinate the motion of one or more floating links with the overall mechanism motion. The methods presented in the paper focus on the use of 2 exact positions plus 2–3 approximate positions. Examples are provided for the solution of rigid-body guidance problems for both four-bar and six-bar linkages. As with many GCP solutions, the graphical solutions presented are well within the capabilities and understanding of both undergraduate students and the practicing engineer.

GPDOF — A FAST ALGORITHM TO DECOMPOSE UNDER-CONSTRAINED GEOMETRIC CONSTRAINT SYSTEMS: APPLICATION TO 3D MODELING

2006 ◽  
Vol 16 (05n06) ◽  
pp. 479-511 ◽  
Author(s):  
GILLES TROMBETTONI ◽  
MARTA WILCZKOWIAK
Keyword(s):  
Equation System ◽  
3D Models ◽  
General Purpose ◽  
Constrained Systems ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Maximum Matching ◽  
Constraint System ◽  
Constraint Systems ◽  
Geometrical Constraints

Our approach exploits a general-purpose decomposition algorithm, called GPDOF, and a dictionary of very efficient solving procedures, called r-methods, based on theorems of geometry. GPDOF decomposes an equation system into a sequence of small subsystems solved by r-methods, and produces a set of input parameters.1. Recursive assembly methods (decomposition-recombination), maximum matching based algorithms, and other famous propagation schema are not well-suited or cannot be easily extended to tackle geometric constraint systems that are under-constrained. In this paper, we show experimentally that, provided that redundant constraints have been removed from the system, GPDOF can quickly decompose large under-constrained systems of geometrical constraints. We have validated our approach by reconstructing, from images, 3D models of buildings using interactively introduced geometrical constraints. Models satisfying the set of linear, bilinear and quadratic geometric constraints are optimized to fit the image information. Our models contain several hundreds of equations. The constraint system is decomposed in a few seconds, and can then be solved in hundredths of seconds.

4MDS: A Geometric Constraint Based Motion Design Software for Synthesis and Simulation of Planar Four-Bar Linkages

2014 ◽  
Author(s):  
Anurag Purwar ◽  
Abhijit Toravi ◽  
Q. J. Ge
Keyword(s):  
Real Time ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Design System ◽  
Dimensional Synthesis ◽  
Complete Control ◽  
Dimensional Changes ◽  
Design Software ◽  
Four Bar Linkage ◽  
Motion Design

This paper presents our recent work on designing and developing a geometric constraint based motion design software system for planar four-bar linkages. Given a motion task, the software computes possible four-bar linkage topologies as well as its dimensions. This capability to analyze the given task and find the best type of the linkage and the dimensions simultaneously sets it apart from any other linkage design software. The Four-Bar Motion Design System (4MDS) makes the synthesis and simulation capabilities available to mechanism designers in an intuitive graphical user interface (GUI) environment. Instead of taking a black box approach to mechanism design, wherein the user simply enters the motion requirements and the software outputs parameters of mechanisms, this software facilitates a dialog with the designer by providing various paths to simulation and synthesis in a design session. The designer has complete control over the specification of motion task, interactive tweaking of the motion, choice of linkage topology computed, dimensional changes, and their apparent effect on motion, all done in real time. This interactivity enhances designers kinematic experience. The underlying theoretical foundation of this paper is based on our earlier work on a task-driven approach to unified type and dimensional synthesis of planar four-bar linkage mechanisms. Instead of treating a planar four-bar mechanism as a set of connected rigid links and joints, we treat them as line or circle constraint generators. With that view, the synthesis problem is reduced to extracting geometric constraints hidden in a given motion task and the simulation is reduced to assembling constraints realizable by mechanical dyads. The algorithm employed is simple and efficient and permits real-time computation, and thus precludes using a limiting database-oriented approach. This tool should make innovation of mechanical motion generating devices accessible to novice and experienced designers alike.

Geometric Constraint-based Reconfiguration and Self-motions of a 4-CRU Parallel Mechanism

2021 ◽  
pp. 1-21
Author(s):  
Latifah Nurahmi ◽  
Pradiktio Putrayudanto ◽  
Guowu Wei ◽  
Sunil K. Agrawal
Keyword(s):  
Parallel Mechanism ◽  
New Technology ◽  
Operation Mode ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Primary Decomposition ◽  
Sustainable Housing ◽  
Additional Mode ◽  
Moving Platform ◽  
Self Motion

Abstract This paper aims to investigate the reconfiguration and self-motions of a 4-CRU parallel mechanism based on the mechanism geometric constraints. The targeted application of such mechanism in this research is for 3D-printing buildings of multi-directional nozzle as a new technology for constructing sustainable housing. By using primary decomposition, four geometric constraints are identified and the reconfiguration analysis is carried out in each of these. It reveals that each geometric constraint will have three distinct operation modes, namely Schoenflies mode, reversed Schoenflies mode and an additional mode. The additional mode can be either 4-DOF mode or it degenerates into 3-DOF mode, depending on the type of the geometric constraint. By taking into account the actuation and constraint singularities, the workspace of each operation mode is analysed and geometrically illustrated. It allows us to determine the regions in which the reconfiguration takes place. Furthermore, the moving-platform can still perform at least 1-DOF self-motion. It occurs at two specific actuated leg lengths. Demonstration of reconfiguration process and self-motions are also provided through a mock-up prototype.

Reconfiguration Techniques and Geometric Constraints of Metamorphic Mechanisms

2009 ◽  
Author(s):  
Liping Zhang ◽  
Jian S. Dai ◽  
Ting-Li Yang
Keyword(s):  
Group Structure ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Geometric Representation ◽  
Group Extensions ◽  
Matrix Operations ◽  
Group Motion ◽  
Displacement Group ◽  
Configuration Changes ◽  
Invariant Configuration

This paper proposes a geometric way to generate metamorphic configurations and investigates metamorphic principles based on geometrized displacement group. Metamorphic reconfiguration techniques are revealed as the variations of kinematic joints, kinematic links and geometric orientation constraints particularly by examining the invariant configuration properties of a mechanism. The nature of all these configuration changes belongs to geometric constraint category. Metamorphic configuration units are proposed as the irreducible reconfiguration modules to envelop these reconfiguration techniques. It can self-reconfigure or be combined to generate metamorphosis. Moreover, the geometrized displacement group is lent to achieve a geometric representation for configuration modelling and further reconfiguration operations. Based on seting up kinematic group extended qualitatively according to its group structure, geometrized displacement group modelling is proposed for these identified metamorphic configuration units. The investigated group motion-matrix is an integration of its displacement group properties and kinematic extensions. Then defined geometric constraint relations and the proposed dependence rules lead to metamorphic principles. In this way, metamorphic process is mapped to matrix operations under group extensions and their compositions. Design examples and a metamorphic joint with six configurations are given to illustrate the feasibility of these metamorphic principles.

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.

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