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.

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.

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.

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.

Computer Simulation of Path and Motion Generation With Six-Bar Linkage

2003 ◽  
Author(s):  
Lu Yi ◽  
Tatu Leinonen
Keyword(s):  
Computer Simulation ◽  
Analytical Calculation ◽  
Geometric Constraint ◽  
Simulation Approach ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Basic Tool ◽  
Kinematic Synthesis ◽  
Four Bar Linkage ◽  
Accuracy And Repeatability

The basic tool of path or motion generation synthesis for more than four prescribed positions is analytical calculation, but its process is quite complicated and far from straightforward. A novel computer simulation mechanism of six-bar linkage for path or motion generation synthesis is presented in this paper. In the case of five-precision points, using the geometric constraint and dimension-driving techniques, a primary simulation mechanism of four-bar linkage is created. Based on the different tasks of path and motion generation for kinematic dimensional synthesis, the simulation mechanisms of path and motion generation with Stephenson I, II and Watt six-bar linkages are developed from the primary simulation mechanism. The results of kinematic synthesis for five prescribed positions prove that the mechanism simulation approach is not only fairly quick and straightforward, but is also advantageous from the viewpoint of accuracy and repeatability.

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.

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.

Mathematical Model for Parametric Tooth Surface of Cylindrical Gears Based on Kinematic Synthesis

2007 ◽  
Author(s):  
Carlos Garci´a-Masia´ ◽  
Juan D. Morillas-A´lvarez
Keyword(s):  
Mathematical Model ◽  
Tooth Surface ◽  
Asymmetry Ratio ◽  
Contact Ratio ◽  
Motion Generation ◽  
Kinematic Synthesis ◽  
Transmission Errors ◽  
Cylindrical Gears ◽  
Tooth Surfaces ◽  
And Function

A generalized approach for parametrizing conjugate tooth surfaces in cylindrical gears is presented in this work. Developed are the polynomials expressions to define the tooth surfaces of pinion and gear based on kinematics synthesis for planar gears. The polynomials expressions incorporate the motion generation (points or positions of precision) and function of transmission errors. It is interesting to note that if the desired pressure angle for the tooth profile is constant, the output polynomial of profile becomes a conventional involute. Polynomials expressions are given for the profile modifications necessary to compensate for any specified or anticipated errors of assembly and/or manufacturing. In addition property of rack as the limits of zone active, transverse contact ratio and contact asymmetry ratio are analysed.

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.

Optimal and Conforming Motion of a Point in a Constrained Plane

1989 ◽  
Author(s):  
C. Ahrikencheikh ◽  
A. A. Seireg ◽  
B. Ravani
Keyword(s):  
Free Space ◽  
Time Complexity ◽  
Velocity Vector ◽  
Automatic Generation ◽  
Geometric Constraints ◽  
Computational Time ◽  
Motion Generation ◽  
Optimal Point ◽  
Generation Algorithm

Abstract This paper deals with automatic generation of motion of a point under both geometric and non-geometric constraints. Optimal point paths are generated which are not only free of collisions with polygonal obstacles representing geometric constraints but also conform to non-geometric constraints such as speed of the motion, a maximum allowable change in the velocity vector and a minimum clearance from the obstacle boundaries. The concept of passage networks and conforming paths on the network are introduced. These are used to provide a new representation of the free space as well as a motion generation algorithm with a computational time complexity of only O(n3.log(n)), where n designates the total number of obstacle vertices. The algorithm finds the shortest or fastest (curved) path that also conforms with preset constraints on the motion of the point. The point paths generated are proved to be optimal while conforming to the constraints.

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