Mixed Exact-Approximate Planar Mechanism Position Synthesis

1977 ◽  
Vol 99 (2) ◽  
pp. 434-439 ◽  
Author(s):  
G. H. Sutherland
Keyword(s):  
Rigid Body ◽  
Digital Computer ◽  
Algebraic Solution ◽  
Numerical Examples ◽  
Planar Mechanism ◽  
Straight Line ◽  
Position Synthesis ◽  
Scientific Calculator ◽  
Exact Positions ◽  
Solution Techniques

Design equations are developed for rigid body guidance through K equals one, two, three, and four exact positions and N – K approximate positions. Both circle and straight-line points are considered. Algebraic solution techniques for use with a pocket scientific calculator and a digital computer are illustrated using numerical examples.

A planar reconfigurable linear rigid-body motion linkage with two operation modes

2014 ◽  
Vol 228 (16) ◽  
pp. 2985-2991 ◽  
Author(s):  
Guangbo Hao ◽  
Xianwen Kong ◽  
Xiuyun He
Keyword(s):  
Rigid Body ◽  
Single Mode ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Reference Guide ◽  
Revolute Joints ◽  
Operation Modes ◽  
Straight Line ◽  
Motion Stage ◽  
Motion Mode

A planar reconfigurable linear (also rectilinear) rigid-body motion linkage (RLRBML) with two operation modes, that is, linear rigid-body motion mode and lockup mode, is presented using only R (revolute) joints. The RLRBML does not require disassembly and external intervention to implement multi-task requirements. It is created via combining a Robert’s linkage and a double parallelogram linkage (with equal lengths of rocker links) arranged in parallel, which can convert a limited circular motion to a linear rigid-body motion without any reference guide way. This linear rigid-body motion is achieved since the double parallelogram linkage can guarantee the translation of the motion stage, and Robert’s linkage ensures the approximate straight line motion of its pivot joint connecting to the double parallelogram linkage. This novel RLRBML is under the linear rigid-body motion mode if the four rocker links in the double parallelogram linkage are not parallel. The motion stage is in the lockup mode if all of the four rocker links in the double parallelogram linkage are kept parallel in a tilted position (but the inner/outer two rocker links are still parallel). In the lockup mode, the motion stage of the RLRBML is prohibited from moving even under power off, but the double parallelogram linkage is still moveable for its own rotation application. It is noted that further RLRBMLs can be obtained from the above RLRBML by replacing Robert’s linkage with any other straight line motion linkage (such as Watt’s linkage). Additionally, a compact RLRBML and two single-mode linear rigid-body motion linkages 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.

The Mechanical Performance Evaluation of a Mechanism with Curved Edge Driving Component

2013 ◽  
Vol 834-836 ◽  
pp. 1290-1294
Author(s):  
Xin Qin Liu
Keyword(s):  
Performance Evaluation ◽  
Fast Moving ◽  
Mechanical Performance ◽  
Gain Coefficient ◽  
The Other ◽  
Force Transmission ◽  
Transmission Performance ◽  
Connecting Rod ◽  
Numerical Examples ◽  
Straight Line

Mechanicalmethods were employed to study the motion and force transmission performance ofa kind of connecting rod slider mechanism with a curved edge driving component.The deduction methods and the computation formulae of the slider displacement,velocity, acceleration and the executive force gain coefficient were given.Considering two cases of the driving components with straight line edge andexponential function edge, the numerical examples was computed respectively,the results show that the former one is suitable for the force transmission andcan be used in the grip design and the other one is suitable for the motiontransmission which can be used in the fast moving mechanism

scholarly journals Optical axes of catadioptric systems including visual, Purkinje and other nonvisual systems of a heterocentric astigmatic eye

2010 ◽  
Vol 69 (3) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Visual System ◽  
Optical Axis ◽  
The Other ◽  
Optical Systems ◽  
Numerical Examples ◽  
Straight Line ◽  
Optical Axes ◽  
Catadioptric Systems ◽  
Reflecting Surfaces ◽  
Special Case

For a dioptric system with elements which may be heterocentric and astigmatic an optical axis has been defined to be a straight line along which a ray both enters and emerges from the system.  Previous work shows that the dioptric system may or may not have an optical axis and that, if it does have one, then that optical axis may or may not be unique.  Formulae were derived for the locations of any optical axes.  The purpose of this paper is to extend those results to allow for reflecting surfaces in the system in addition to refracting elements.  Thus the paper locates any optical axes in catadioptric systems (including dioptric systems as a special case).  The reflecting surfaces may be astigmatic and decentred or tilted.  The theory is illustrated by means of numerical examples.  The locations of the optical axes are calculated for seven optical systems associated with a particular heterocentric astigmatic model eye.  The optical systems are the visual system, the four Purkinje systems and two other nonvisual systems of the eye.  The Purkinje systems each have an infinity of optical axes whereas the other nonvisual systems, and the visual system, each have a unique optical axis. (S Afr Optom 2010 69(3) 152-160)

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.

Two Precision Position Synthesis of Planar Linkages With Positional Rectification

1996 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Numerical Algorithms ◽  
All Solutions ◽  
Pass Through ◽  
Planar Linkages ◽  
Position Synthesis ◽  
Exact Positions

Abstract Precision position synthesis is used to generate planar linkages that pass through two exact positions and an additional number of approximate positions. The approximate positions provide a means of rectifying the solution linkages such that all solutions presented are more likely to describe a motion that remains within acceptable positional bounds. The rectification method involves the development of three different numerical algorithms that may be applied to a particular step in the dyad/triad method of precision position synthesis. The three algorithms presented can be applied in a variety of combinations to allow for the synthesis of both simple (four-bar) and complex (multiloop) planar linkages.

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.

Three-Position Rigid Body Guidance Using Specified Moving Pivots for a Four-Bar Mechanism With Variable Topology

2018 ◽  
Author(s):  
Brian J. Slaboch
Keyword(s):  
Rigid Body ◽  
Algebraic Solution ◽  
Graphical Approach ◽  
Practical Applications ◽  
Joint Geometry ◽  
Topological State ◽  
Variable Topology

This paper provides an algorithm allowing a designer to perform three position rigid body guidance with specified moving pivots for a 4R-RRRP mechanism with variable topology (MVT). A mechanism with variable topology is a mechanism that changes from one topological state to another due to a change in joint geometry. Both a graphical approach and an algebraic solution are presented. An example is provided in which a circuit defect in a 4R mechanism can be avoided using a 4R-RRRP mechanism. Two additional examples are provided that show the results of this new theory. Practical applications for this theory are found in many industries including manufacturing, aerospace, and healthcare.

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.

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