Using Optimization for the Mixed Exact-Approximate Synthesis of Planar Mechanisms

2015 ◽  
Author(s):  
Jugesh Sundram ◽  
Pierre Larochelle
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Parameter Space ◽  
Design Parameter ◽  
Linear Optimization ◽  
Planar Mechanisms ◽  
Synthesis Algorithm ◽  
Approximate Synthesis ◽  
Position Synthesis ◽  
Novel Algorithm

Exact synthesis algorithms for planar mechanisms for rigid-body guidance are limited by the number of poses the mechanism can position the rigid-body in Euclidean space. The mixed exact-approximate synthesis algorithm described guides a rigid exactly through two positions and approximately through n guiding positions. It breaks down a rigid-body guidance task into n sub-problems of three positions to be solved by an exact synthesis algorithm. A novel algorithm utilizing MATLAB’s constrained non-linear optimization tools is proposed. The algorithm can be utilized to find within a bounded design parameter space the RR dyad that exactly reaches two positions and minimizes the distance to n positions. Two such dyads can be synthesized independently and then combined to yield a planar four-bar mechanism. An example using the proposed algorithm to design a planar four-bar mechanism to solve McCarthy’s 11-position synthesis problem stated at the 2002 ASME Conference is included.

Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)

1994 ◽  
Author(s):  
Pierre Larochelle ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Numerical Results ◽  
Image Space ◽  
Dimensional Synthesis ◽  
Invariant Metric ◽  
Planar Mechanisms ◽  
Position And Orientation ◽  
Position Synthesis

Abstract In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.

On the Seven Position Synthesis of a 5-SS Platform Linkage

1997 ◽  
Vol 123 (1) ◽  
pp. 74-79 ◽  
Author(s):  
Qizheng Liao ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Reference Point ◽  
Polynomial Solution ◽  
Degree Of Freedom ◽  
Prismatic Joint ◽  
Movement Characteristics ◽  
Position Synthesis ◽  
Freedom Movement

This paper builds on Innocenti’s polynomial solution for the 5-SS platform that generates a one-degree of freedom movement through seven specified spatial positions of a rigid body. We show that his 60×60 resultant can be reduced to one that is 10×10. We then actuate the linkage using a prismatic joint on the sixth leg and determine the trajectory of the reference point through the specified positions. The singularity submanifold of this associated 6-SS platform provides information about the movement characteristics of the 5-SS linkage.

Five Position Synthesis of Spatial CC Dyads

1994 ◽  
Author(s):  
Andrew P. Murray ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Kinematic Chain ◽  
New Technique ◽  
Screw Axis ◽  
Fixed Axis ◽  
A New Technique ◽  
Position Synthesis

Abstract This paper presents a new technique for determining the fixed axes of spatial CC dyads for rigid body guidance through five finitely separated positions. A CC dyad is a kinematic chain consisting of a floating link connected by a cylindric joint to a crank which in turn is connected to ground by a second cylindric joint. The lines that can be axes of the fixed joint are shown to be obtained from a “compatibility platform” constructed from selected relative screw axes associated with the five specified displacements. We show that the screw axis of the displacement of this platform is a fixed axis of a CC dyad compatible with the five positions. Roth’s original example is presented to verify the calculations. The specialization of this procedure to planar and spherical five position synthesis is also presented.

Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad

2012 ◽  
Author(s):  
Chintien Huang ◽  
Chenning Hung ◽  
Kuenming Tien
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Quadratic Equation ◽  
Geometric Constraints ◽  
Numerical Continuation ◽  
Polynomial Equations ◽  
Quartic Polynomial ◽  
Solutions Of Equations ◽  
Position Synthesis

This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.

A New Approach for Exact/Approximate Point Synthesis of Planar Mechanisms

2005 ◽  
Author(s):  
Ahmad Smaili ◽  
Nadim Diab
Keyword(s):  
Synthesis Method ◽  
Dimensional Synthesis ◽  
Gradient Search ◽  
Planar Mechanisms ◽  
Simple Method ◽  
New Approach ◽  
Optimum Synthesis ◽  
Approximate Synthesis ◽  
Optimum Solution ◽  
Synthesis Approach

The aim of this article is to provide a simple method to solve the mixed exact-approximate dimensional synthesis problem of planar mechanism. The method results in a mechanism that can traverse a closed path with the choice of any number of exact points while the rest are approximate points. The algorithm is based on optimum synthesis rather than on precision position methods. Ant-gradient search is applied on an objective function based on log10 of the error between the desired positions and those generated by the optimum solution. The log10 function discriminates on the side of generating miniscule errors (on the order of 10−14) at the exact points while allowing for higher errors at the approximate positions. The algorithm is tested by way of five examples. One of these examples was used to test exact/approximate synthesis method based on precision point synthesis approach.

Synthesis of Planar Mechanisms for Pick and Place Tasks With Guiding Locations

2013 ◽  
Author(s):  
Pierre Larochelle
Keyword(s):  
Algebraic Geometry ◽  
Constraint Equation ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Open Chain ◽  
Synthesis Algorithm ◽  
Kinematic Chains ◽  
Synthesis Technique ◽  
Kinematic Inversion

A novel dimensional synthesis technique for solving the mixed exact and approximate motion synthesis problem for planar RR kinematic chains is presented. The methodology uses an analytic representation of the planar RR dyads rigid body constraint equation in combination with an algebraic geometry formulation of the exact synthesis for three prescribed locations to yield designs that exactly reach the prescribed pick & place locations while approximating an arbitrary number of guiding locations. The result is a dimensional synthesis technique for mixed exact and approximate motion generation for planar RR dyads. A solution dyad may be directly implemented as a 2R open chain or two solution dyads may be combined to form a planar 4R closed chain; also known as a planar four-bar mechanism. The synthesis algorithm utilizes only algebraic geometry and does not require the use of a numerical optimization algorithm or a metric on planar displacements. Two implementations of the synthesis algorithm are presented; computational and graphical construction. Moreover, the kinematic inversion of the algorithm is also included. An example that demonstrates the synthesis technique is included.

Finite Position Synthesis of 5-SS Platform Linkages Including Partially Specified Joint Locations

2017 ◽  
Author(s):  
Xin Ge ◽  
Anurag Purwar ◽  
Q. J. Ge
Keyword(s):  
Linear Structure ◽  
Motion Generation ◽  
Numerical Examples ◽  
Moving Points ◽  
Unified Algorithm ◽  
Moving Platform ◽  
Position Synthesis ◽  
Partial Specification ◽  
Generation Problem ◽  
Novel Algorithm

A 5-SS platform linkage generates a one-degree-of-freedom motion of a moving platform such that each of five moving points on the platform is constrained on a sphere, or in its degenerated case, on a plane. It has been well established a 5-SS platform linkage can be made to guide though seven positions exactly. This paper investigates the cases when the number of given positions are less than seven that allows for partial specification of locations of the moving points. A recently developed novel algorithm with linear structure in the design equations has been extended for the solution of the problem. The formulation of this expanded motion generation problem unifies the treatment of the input positions and constraints on the moving and fixed joints associated with the 5-SS platform linkage. Numerical examples are provided to show the effectiveness of the unified algorithm.

Approximate Velocities in Mixed Exact-Approximate Position Synthesis of Planar Mechanisms

1999 ◽  
Vol 123 (3) ◽  
pp. 388-394 ◽  
Author(s):  
Jennifer E. Holte ◽  
Thomas R. Chase ◽  
Arthur G. Erdman
Keyword(s):  
Two Dimensional ◽  
Computer Implementation ◽  
Motion Generation ◽  
Planar Mechanisms ◽  
Form Complex ◽  
New Approach ◽  
Realistic Representation ◽  
Velocity Constraints ◽  
Position Synthesis ◽  
Exact Positions

A new approach to the synthesis of planar linkage mechanisms with approximate velocity constraints is proposed. The paper presents the first closed-form complex-number dyad solution to the ground pivot specification problem for two precision positions with velocity specified at one of the positions. The solution is then manipulated in order to add approximate velocity constraints to design methods for two exact positions and an unlimited number of approximate positions. The approximate position and velocity constraints facilitate more realistic representation of design objectives. Solution spaces are presented using two-dimensional ground-pivot maps. Computer implementation of the proposed methodologies would allow designers with little or no knowledge of the synthesis techniques to interactively explore maps of solutions for four-bar motion generation.

Synthesis of Four-Link Space Mechanisms Via Extension of Point-Position-Reduction Technique

1971 ◽  
Vol 93 (1) ◽  
pp. 85-89 ◽  
Author(s):  
A. H. Soni ◽  
Matthew Huang
Keyword(s):  
Reduction Technique ◽  
Planar Mechanisms ◽  
Point Position ◽  
Position Synthesis ◽  
Space Mechanisms

The principle of point-position-reduction technique, which is used for position-synthesis of planar mechanisms, is extended to synthesize spherical four-link and spatial four-link RCCC mechanisms for four precision positions.

scholarly journals Rigid Body Trajectories in Different 6D Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Carol Linton ◽  
William Holderbaum ◽  
James Biggs
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Hyperbolic Space ◽  
Poisson Structure ◽  
Scaling Factor ◽  
The Body ◽  
Body Frame ◽  
Spatial Frame ◽  
Axis Of Symmetry ◽  
Rotational Motions

The objective of this paper is to show that the group with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.

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