On the Input Joint Rotation Space and Mobility of Linkages

2008 ◽  
Vol 130 (9) ◽  
Author(s):  
Kwun-Lon Ting
Keyword(s):  
Visualization Tool ◽  
Mobility Analysis ◽  
Joint Rotation ◽  
Single Loop ◽  
Spatial Linkages ◽  
One To One ◽  
Spherical Linkages ◽  
Virtual Loop

This paper presents the concept and application of input joint rotation space of linkages and offers updates on the N-bar rotatability laws. A thorough discussion on the joint rotation space of single-loop planar five-bar linkages is first presented. The concept is then extended to spherical linkages and the generalization to N-bar linkages is discussed. It offers a visualization tool for the input joint rotatability and fills up a void in the N-bar rotatability laws regarding the coordination among multiple inputs. It explains the formation of branches and how to establish a one-to-one correspondence between the inputs and the linkage configurations. The applications to multiloop linkages and spatial linkages are highlighted with Stephenson six-bar linkages, geared linkages, and spatial RCRCR mechanisms. These examples exhibit simplicity and benefits of the proposed concept to the mobility analysis of diversified mechanisms. The concept of virtual loop in spatial linkages is proposed and demonstrated with simple RCRCR and Stephenson six-bar mechanisms.

Joint Rotation Space and Mobility of Linkages

2007 ◽  
Author(s):  
Kwun-Lon Ting
Keyword(s):  
Fundamental Principle ◽  
General Concept ◽  
Joint Rotation ◽  
Single Loop ◽  
Input Domain ◽  
Spatial Linkages ◽  
One To One

The paper presents a general concept regarding the joint rotation space (JRS) of single loop planar and spherical N-bar linkages as well as its significance and applicability in multiple loop and spatial linkages. A JRS refers to the input domain of a linkage. The paper first discusses and classifies the JRS types of single loop five-bar linkages based on the types of the JRS boundary. The concept of sheets and pages of JRS is then introduced as an aid to understand the mobility of linkages. A sheet refers to the JRS of a linkage branch (or circuit). A JRS sheet may have one or more pages and each page refers to an uncertainty singularity free JRS. The concept is then generalized to any single loop planar and spherical N-bar linkages (N ≥ 3). The paper offers geometric insights to the input domain of a linkage and establishes a one-to-one correspondence between the input domain and the linkage configurations. The applications to the mobility identification of complex linkages are discussed and demonstrated with geared five-bar linkages, Stephenson six-bar linkages, as well as spatial RCRCR mechanisms. The paper presents a useful model explaining the fundamental principle regarding the formation of branches and sub-branches in Stephenson six-bar linkages as well as some other multiloop and spatial linkages. The Mobility similarity between multiloop linkages and spatial linkages is also highlighted.

Branch Identification of Spherical Six-Bar Linkages

2016 ◽  
Author(s):  
Liangyi Nie ◽  
Jun Wang ◽  
Kwun-Lon Ting ◽  
Daxing Zhao ◽  
Quan Wang ◽  
...  
Keyword(s):  
Range Of Motion ◽  
Degree Of Freedom ◽  
Joint Rotation ◽  
Single Degree Of Freedom ◽  
Complex Issue ◽  
Single Degree ◽  
Identification Method ◽  
Spatial Linkages ◽  
Spherical Linkages

Branch (assembly mode or circuit) identification is a way to assure motion continuity among discrete linkage positions. Branch problem is the most fundamental, pivotal, and complex issue among the mobility problems that may also include sub-branch (singularity-free) identification, range of motion, and order of motion. Branch and mobility complexity increases greatly in spherical or spatial linkages. This paper presents the branch identification method suitable for automated motion continuity rectification of a single degree-of-freedom of spherical linkages. Using discriminant method and the concept of joint rotation space (JRS), the branch of a spherical linkage can be easily identified. The proposed method is general and conceptually straightforward. It can be applied for all linkage inversions. Examples are employed to illustrate the proposed method.

Velocity, Acceleration, and Static-Force Analyses of Spatial Linkages

1965 ◽  
Vol 32 (4) ◽  
pp. 903-910 ◽  
Author(s):  
J. Denavit ◽  
R. S. Hartenberg ◽  
R. Razi ◽  
J. J. Uicker
Keyword(s):  
Virtual Work ◽  
Algebraic Method ◽  
Degree Of Freedom ◽  
Static Force ◽  
Virtual Displacement ◽  
Single Loop ◽  
Loop Equation ◽  
The Matrix ◽  
Spatial Linkages

The algebraic method using 4 × 4 matrices is extended to the analysis of velocities, accelerations, and static forces in one-degree-of-freedom, single-loop, spatial linkages consisting of revolute and prismatic pairs, either singly or in combination. The methods are well suited for machine calculations and have been tested on a number of examples, one of which is presented. Velocities and accelerations are obtained by differentiation of the matrix-loop or position equation. Static forces are found by combining the method of virtual work with the matrix-loop equation to relate the virtual displacement of the load to given virtual deformations of the links.

Unified Mobility Identification and Rectification of Six-Bar Linkages

2009 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Jun Wang ◽  
Changyu Xue
Keyword(s):  
Special Form ◽  
Simple Explanation ◽  
Input Output ◽  
Mobility Analysis ◽  
Joint Rotation ◽  
Input Condition ◽  
Computer Aided ◽  
Unified Method

This paper offers a unified method for a complete and unified treatment on the mobility identification and rectification of any planar and spherical six-bar linkages regardless the linkage type and the choice of the input, output, or fixed links. The method is based on how the joint rotation spaces of the four-bar loop and a five-bar loop in a Stephenson six-bar linkage interact each other. A Watt six-bar linkage is regarded as a special form of Stephenson six-bar linkage via the stretch and rotation of a four-bar loop. The paper offers simple explanation and geometric insights for the formation of branch (circuit), sub-branch, and order of motion of six-bar linkages. All typical mobility issues, including branch, sub-branch, and type of motion under any input condition can be identified and rectified with the proposed method. The method is suitable for automated computer-aided mobility identification. The applicability of the results to the mobility analysis of serially connected multiloop linkages is also discussed.

Mobility of Single Loop Linkages: A Final Word?

2007 ◽  
Author(s):  
Jose´ Mari´a Rico ◽  
J. Jesu´s Cervantes ◽  
Juan Rocha ◽  
Jaime Gallardo ◽  
Luis Daniel Aguilera ◽  
...  
Keyword(s):  
Mobility Analysis ◽  
Additional Insight ◽  
Final Word ◽  
Single Loop ◽  
Kinematic Chains ◽  
Recent Developments ◽  
The Subject ◽  
Insight Into

Setting aside paradoxical linkages such as Bennett’s, Bricard’s or Goldberg’s, the mobility of single loop linkages seemed, with the developments on mobility analysis carried out in the last five years, a closed chapter in kinematic research. However, recent developments on the mobility of parallel platforms have shed additional insight into the problem. This contribution attempts to unify the results obtained in the last five years in the area of mobility of single-loop kinematic chains to state what appears to be a final word on the subject.

General Formula of Mobility Calculation for Planar Mechanism

2012 ◽  
Vol 562-564 ◽  
pp. 654-659 ◽  
Author(s):  
Yan Dong Yang ◽  
Yi Tong Zhang
Keyword(s):  
Calculation Method ◽  
Mobility Analysis ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Link Group ◽  
New Concepts ◽  
General Constraints ◽  
Loop Constraint ◽  
New Formula ◽  
Virtual Loop

Deficiencies are existed for currently formulas of mobility calculation for planar mechanism. They are not suitable for planar mechanism with virtual constraints and the number of general constraints equal to 4. To solve the problem, the new concepts of virtual loop, virtual-loop constraint and virtual pair are defined to establish a general f ormula for DOF of planar mechanism; the calculation method for virtual-loop constraint and the mobility of link-group are also given. It is proved that the new formula is correct, general, simple and effective through the mobility analysis of several different kinds of planar mechanisms.

Rigid Foldability of Triangle-Twist Origami Pattern and its Derived 6R Linkage

2017 ◽  
Author(s):  
Rui Peng ◽  
Jiayao Ma ◽  
Yan Chen
Keyword(s):  
Theoretical Method ◽  
Geometric Parameters ◽  
Degree Of Freedom ◽  
Space Structures ◽  
Engineering Applications ◽  
Systematic Method ◽  
Mountain Valley ◽  
Folding Pattern ◽  
Spatial Linkages ◽  
Spherical Linkages

Rigid origami is an important subset of origami with broad engineering applications from space structures to metamaterials. The rigid foldability of an origami pattern is determined by both the geometric parameters and the mountain-valley crease assignment. In this paper, by using the equivalent relationships between origami vertices and spherical linkages, a systematic method was proposed to analyze the motion of the triangle-twist pattern with varying distribution of mountain and valley creases, and its rigid folding types were identified. Moreover, kirigami technology was applied to the rigid folding pattern without changing its degree of freedom, from which a new kind of overconstrained 6R linkage was developed. The theoretical method proposed in this paper can be readily extended to study other types of origami patterns, which will in turn help to design structures with large deployable ratio as well as some new spatial linkages.

On the Mobility of Spatial Group 2 Mechanisms

2009 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Changyu Xue ◽  
Jun Wang ◽  
Kenneth R. Currie
Keyword(s):  
Branch Points ◽  
Mobility Analysis ◽  
Spatial Linkages ◽  
Group 2 ◽  
Fundamental Equations

Spatial linkages are classified into four groups according to the number of fundamental equations or virtual loops that govern linkage displacement. The number of virtual loops represents the complexity of a spatial linkage as that of planar or spherical multiloop linkages. The concept of generalized branch points offers the explanation of how branches are formed in spatial group 2 linkages. In this paper, the mobility analysis is carried out based on the similarity of the mobility features rather than the specific or individual linkage structure. A branch rectification scheme is presented and demonstrated with examples.

On the Rotatability of Spherical N-Bar Chains

1994 ◽  
Vol 116 (3) ◽  
pp. 920-923 ◽  
Author(s):  
Yung-Way Liu ◽  
Kwun-Lon Ting
Keyword(s):  
Degree Of Freedom ◽  
Spatial Linkages ◽  
Spherical Linkages ◽  
Planar Linkages

This paper established the spherical counterparts of Ting’s rotatability laws of planar linkages. Generally speaking, similar rotatability properties exist between a pair of adjoining links in planar and spherical linkages and the concept of invariant link rotatability is valid for spherical linkage only if it is referred to that between adjoining links. The established spherical rotatability criteria are also valid for single and multiple degree of freedom nR3C, nR2C1P, nR1C2P and nR3P spatial linkages.

A New Approach for the Orders of the Bennett-Based Linkages in Mobility Analysis

2009 ◽  
Author(s):  
Jingfang Liu ◽  
Zhen Huang ◽  
Yanwen Li
Keyword(s):  
Superposition Principle ◽  
Mobility Analysis ◽  
Linear Superposition ◽  
Geometric Properties ◽  
New Approach ◽  
Single Loop ◽  
Linear Superposition Principle ◽  
Overconstrained Linkages

Among the 3D single-loop overconstrained linkages, quite a number of them are combinations of Bennett linkages. Mobility on the overconstrained linkages including the Bennett-based linkages is known to be one of the difficult topics in kinematics. In the paper, a new approach based on the linear superposition principle for determining the orders of Bennett-based linkages is proposed, and the mobility of some typical Bennett-based linkages is calculated with the Modified Gru¨bler-Kutzbach criterion. In addition, geometric properties of some of the screw systems are employed to identify whether the mobility is global.

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