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.

Kinematic Synthesis for Infinitesimally and Multiply Separated Positions Using Geometric Constraint Programming

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
James P. Schmiedeler ◽  
Barrett C. Clark ◽  
Edward C. Kinzel ◽  
Gordon R. Pennock
Keyword(s):  
Undergraduate Students ◽  
Constraint Programming ◽  
Computer Aided Design ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Kinematic Synthesis ◽  
Design Software ◽  
Aided Design

Geometric constraint programming (GCP) is an approach to synthesizing planar mechanisms in the sketching mode of commercial parametric computer-aided design software by imposing geometric constraints using the software's existing graphical user interface. GCP complements the accuracy of analytical methods with the intuition developed from graphical methods. Its applicability to motion generation, function generation, and path generation for finitely separated positions has been previously reported. By implementing existing, well-known theory, this technical brief demonstrates how GCP can be applied to kinematic synthesis for motion generation involving infinitesimally and multiply separated positions. For these cases, the graphically imposed geometric constraints alone will in general not provide a solution, so the designer must parametrically relate dimensions of entities within the graphical construction to achieve designs that automatically update when a defining parameter is altered. For three infinitesimally separated positions, the designer constructs an acceleration polygon to locate the inflection circle defined by the desired motion state. With the inflection circle in place, the designer can rapidly explore the design space using the graphical second Bobillier construction. For multiply separated position problems in which only two infinitesimally separated positions are considered, the designer constrains the instant center of the mechanism to be in the desired location. For example, four-bar linkages are designed using these techniques with three infinitesimally separated positions and two different combinations of four multiply separated positions. The ease of implementing the techniques may make synthesis for infinitesimally and multiply separated positions more accessible to mechanism designers and undergraduate students.

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.

Function Generation With Finitely Separated Precision Points Using Geometric Constraint Programming

2006 ◽  
Vol 129 (11) ◽  
pp. 1185-1190 ◽  
Author(s):  
Edward C. Kinzel ◽  
James P. Schmiedeler ◽  
Gordon R. Pennock
Keyword(s):  
Nonlinear Equations ◽  
Constraint Programming ◽  
Parametric Design ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Function Generation ◽  
Large Numbers ◽  
Design Software ◽  
Complex Mechanisms ◽  
Four Bar Linkage

This paper extends geometric constraint programming (GCP) to function generation problems involving large numbers of finitely separated precision points and complex mechanisms. In parametric design software, GCP uses the sketching mode to graphically impose geometric constraints in kinematic diagrams and the numerical solvers to solve the relevant nonlinear equations without the user explicitly formulating them. For function generation, the same approach can be applied to any mechanism, requiring no unique algorithms. Implementation is straightforward, so the designer can quickly generate solutions for a large number of precision points and/or with complex mechanisms to accurately match the function. Examples of function generation with a four-bar linkage, a Stephenson III six-bar linkage, and a seven-bar linkage with a mobility of two are presented.

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.

A GCP Design Approach for Infinitesimally and Multiply Separated Positions Based on the Use of Velocity and Acceleration Vector Diagrams

2014 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Design Process ◽  
Constraint Programming ◽  
Parametric Modeling ◽  
Geometric Constraint ◽  
Design Approach ◽  
Powerful Method ◽  
Planar Mechanisms ◽  
Acceleration Vector ◽  
Acceleration Analysis ◽  
Four Bar Linkage

Geometric Constraint Programming (GCP) provides a powerful method for the synthesis of planar mechanisms using parametric modeling programs that are common to industry. The graphical nature of GCP allows for the ready incorporation of many existing graphical constructions into the design process. This paper examines the use of vector diagrams for velocity and acceleration analysis of a four-bar linkage and how such diagrams can be incorporated into the design process using the methods of GCP. The method is implemented by using GCP to create a mechanism at one or more design positions. Velocity and acceleration vector diagrams are added to positions of interest to allow for the inclusion of velocity and acceleration information in the design process. The result is an approach to GCP synthesis that allows a designer to create mechanisms to match requirements for infinitesimally and multiply separated positions using techniques that are commonly taught in an introductory undergraduate mechanisms course. Two examples are presented to demonstrate the utility of the methods described.

Using Modeling Software to Visualize Biomechanics in an Introductory Undergraduate Bioengineering Course

2012 ◽  
Author(s):  
Stephanie M. Kusano
Keyword(s):  
Undergraduate Students ◽  
Human Body ◽  
Visual Tool ◽  
Human Body Modeling ◽  
Industry Standard ◽  
Wide Range ◽  
Basic Understanding ◽  
Modeling Software ◽  
Second Year ◽  
Crash Response

The modeling software MADYMO was used as a visual teaching tool for teaching human body modeling during an introductory undergraduate bioengineering course. The human body modeling lecture was covered during the six-week biomechanics section of a 15-week course. The students in this introductory bioengineering course were primarily second year, second semester students, from a wide range of engineering disciplines. Biomechanics is a challenging topic to teach to undergraduate students because of the students’ limited understanding of general mechanics and physiology. Education research has shown the effectiveness of using visual teaching tools (Felder, 1988). MADYMO is an industry-standard modeling tool for human crash response. Considering the challenges of introducing human body modeling in a single lecture to undergraduate students, we believe using MADYMO as a visual tool helps the students realize a basic understanding and appreciation for human body modeling.

Kinematic Synthesis for Infinitesimally and Multiply Separated Positions Using Geometric Constraint Programming

2011 ◽  
Author(s):  
James P. Schmiedeler ◽  
Barrett C. Clark ◽  
Edward C. Kinzel ◽  
Gordon R. Pennock
Keyword(s):  
Constraint Programming ◽  
Computer Aided Design ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Kinematic Synthesis ◽  
Function Generation ◽  
Design Software ◽  
Aided Design

Geometric Constraint Programming (GCP) is an approach to synthesizing planar mechanisms in the sketching mode of commercial parametric computer-aided design software by imposing geometric constraints using the software’s existing graphical user interface. GCP complements the accuracy of analytical methods with the intuition developed from graphical methods. Its applicability to motion generation, function generation, and path generation for finitely separated positions has been previously reported. This paper demonstrates how GCP can be applied to kinematic synthesis for motion generation involving infinitesimally and multiply separated positions. For these cases, the graphically imposed geometric constraints alone will in general not provide a solution, so the designer must parametrically relate dimensions of entities within the graphical construction to achieve designs that automatically update when a defining parameter is altered. For three infinitesimally separated positions, the designer constructs an acceleration polygon to locate the inflection circle defined by the desired motion state. With the inflection circle in place, the designer can rapidly explore the design space using the graphical second Bobillier construction. For multiply separated position problems in which only two infinitesimally separated positions are considered, the designer constrains the instant center of the mechanism to be in the desired location. Example four-bar linkages are designed using these techniques with three infinitesimally separated positions and two different combinations of four multiply separated positions.

The Application of Geometric Constraint Programming to the Design of Stephenson III Dwell Linkages

2016 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Constraint Programming ◽  
Angular Displacement ◽  
Geometric Constraint ◽  
Link Length ◽  
Modeling Software ◽  
Dwell Position ◽  
Angular Displacements ◽  
Four Bar Linkage ◽  
Additional Constraints ◽  
Complete Specification

Stephenson III linkages provide a means to create an approximate dwell mechanism without the use of cams. The dwell cycle is created by first choosing or designing a four-bar linkage that contains a coupler path with a near circular segment. An external dyad is attached to the coupler point such that the center of the floating link of the dyad coincides with the center of the circular portion of the coupler curve. This connection produces a dwell in the external dyad as the four-bar linkage traverses the circular portion of the coupler curve. This paper demonstrates how the necessary conditions for a dwell linkage can be obtained with the use of Geometric Constraint Programming (GCP). The construction process is initiated by using GCP techniques to develop a four-bar linkage with a minimum of four path points that lie on a prescribed arc. This part of the problem also uses GCP to apply additional constraints to the four-bar linkage. These include the application of appropriate link dimensions to achieve a Grashof linkage with a crank input, and the specification of the required crank rotation angle during the dwell cycle of the mechanism. Once the four-bar is defined, an external dyad is attached to the coupler link of the four-bar to produce the specified dwell characteristics. The dwell dyad may include for its output either a rotational link whose range of angular travel is defined, or a sliding link whose range of linear motion is defined. GCP techniques are used to enforce a specified range of motion for the output dyad through the use of an instant center construction to define the limits of travel of the four-bar coupler curve relative to the dwell ground pivot. If the dwell dyad is designed for angular displacements, the construction is completed by using GCP to define the desired angular displacement of the dwell link, resulting in the specification of the link length and ground pivot location. If the dwell dyad is a linear (slider) output, the final part of the GCP construction is used to define the desired length of linear travel, which results in the complete specification of the slider path and angle. The GCP techniques are presented with the development of an example, with sample results presented for a dwell mechanism with a rotational dwell cycle, and also for a dwell mechanism with a linear (slider) dwell output. The example demonstrates the ability of GCP methods to use standard solid-modeling software to obtain Stephenson III linkages with dwells that deviate from the dwell position by less than 0.1% of total motion.

Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming

2005 ◽  
Vol 128 (5) ◽  
pp. 1070-1079 ◽  
Author(s):  
Edward C. Kinzel ◽  
James P. Schmiedeler ◽  
Gordon R. Pennock
Keyword(s):  
Constraint Programming ◽  
Computer Aided Design ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Motion Generation ◽  
Robust Algorithms ◽  
Kinematic Synthesis ◽  
Synthesis Techniques ◽  
And Function ◽  
Accuracy And Repeatability

This paper presents an original approach to the kinematic synthesis of planar mechanisms for finitely separated positions. The technique, referred to here as geometric constraint programming, uses the sketching mode of commercial parametric computer-aided design software to create kinematic diagrams. The elements of these diagrams are parametrically related so that when a parameter is changed, the design is modified automatically. Geometric constraints are imposed graphically through a well-designed user interface, and numerical solvers integrated into the software solve the relevant systems of equations without the user explicitly formulating those equations. This allows robust algorithms for the kinematic synthesis of a wide variety of mechanisms to be “programmed” in a straightforward, intuitive manner. The results provided by geometric constraint programming exhibit the accuracy and repeatability achieved with analytical synthesis techniques, while simultaneously providing the geometric insight developed with graphical synthesis techniques. The key advantages of geometric constraint programming are that it is applicable to a broad range of kinematic synthesis problems, user friendly, and highly accessible. To demonstrate the utility of the technique, this paper applies geometric constraint programming to three examples of the kinematic synthesis of planar four-bar linkages: Motion generation for five finitely separated positions, path generation for nine finitely separated precision points, and function generation for four finitely separated positions.

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