Mobility Analysis of Limited-Degrees-of-Freedom Parallel Mechanisms in the Framework of Geometric Algebra

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Qinchuan Li ◽  
Xinxue Chai ◽  
Ji'nan Xiang
Keyword(s):  
Basic Property ◽  
Geometric Algebra ◽  
Degrees Of Freedom ◽  
Linear Equations ◽  
Closed Form Expression ◽  
Parallel Mechanisms ◽  
Motion Pattern ◽  
Mobility Analysis ◽  
Form Expression ◽  
General Method

Mobility is a basic property of a mechanism. The aim of mobility analysis is to determine the number of degrees-of-freedom (DOF) and the motion pattern of a mechanism. The existing methods for mobility analysis have some drawbacks when being applied to limited-DOF parallel mechanisms (PMs). Particularly, it is difficult to obtain a symbolic or closed-form expression of mobility and its geometric interpretations are not always straightforward. This paper presents a general method for mobility analysis of limited-DOF PMs in the framework of geometric algebra. The motion space and constraint space of each limb are expressed using geometric algebra. Then the mobility of the PM can be calculated based on the orthogonal complement relationship between the motion space and the constraint space. The detailed mobility analyses of a 3-RPS PM and a 3-RPC PM are presented. It is shown that this method can obtain a symbolic expression of mobility with straightforward geometric interpretations and is applicable to limited-DOF PMs with or without redundant constraints. Without solving complicated symbolic linear equations, this method also has computational advantages.

Jacobian Derivation of 5-DOF 3R2T Parallel Mechanisms

2004 ◽  
Author(s):  
Qinchuan Li ◽  
Xudong Hu ◽  
Zhen Huang
Keyword(s):  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Screw Theory ◽  
Kinematic Chain ◽  
Parallel Mechanisms ◽  
Mobility Analysis

This paper presents a method for the Jacobian derivation of 5-DOF 3R2T PMs (parallel mechanisms), where 3R denotes three rotational DOFs (degrees of freedom) and 2T denotes two translational DOFs. First the mobility analysis of such kind of parallel mechanisms is reviewed briefly. The Jacobian matrix of the single limb kinematic chain is obtained via screw theory, which is a 6 × 5 matrix. Then it is shown that the mobility analysis of such kind of PM is important when simplifying the 6 × 5 matrix into a 5 × 5 Jacobian matrix. After obtaining the 5 × 5 Jacobian matrix for each limb, a 5 × 5 Jacobian matrix for the whole mechanism can be established.

Type Synthesis of Parallel Mechanisms With Multiple Operation Modes

2006 ◽  
Author(s):  
Xianwen Kong ◽  
Cle´ment M. Gosselin ◽  
Pierre-Luc Richard
Keyword(s):  
Parallel Mechanism ◽  
Translational Motion ◽  
Parallel Mechanisms ◽  
Motion Pattern ◽  
Type Synthesis ◽  
Motion Patterns ◽  
Operation Modes ◽  
Multiple Operation ◽  
Spherical Motion ◽  
General Method

There are usually several motion patterns having the same DOF (degree of freedom). For example, planar motion, spherical motion, and spatial translation are motion patterns with 3-DOF. An f-DOF parallel mechanism with multiple operation modes is a parallel mechanism that can generate different motion patterns with f DOF. Up to now, no method has been proposed for the type synthesis of parallel mechanisms with multiple operation modes. This paper presents a general method for the type synthesis of parallel mechanisms with multiple operation modes. Using the proposed approach, 3-DOF parallel mechanisms with both spherical and translational modes, i.e., parallel mechanisms generating both the spherical motion pattern and the spatial translational motion pattern, are generated systematically. A large number of parallel mechanisms with both spherical and translational modes are obtained.

Error Space Estimation of Three Degrees of Freedom Planar Parallel Mechanisms

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Jianzhong Ding ◽  
Shengnan Lyu ◽  
Ting Da ◽  
Chunjie Wang ◽  
Gregory S. Chirikjian
Keyword(s):  
Degrees Of Freedom ◽  
Geometric Error ◽  
Closed Form Expression ◽  
Parallel Mechanisms ◽  
Analysis Method ◽  
Interval Analysis Method ◽  
Minimum Volume Ellipsoid ◽  
Error Space ◽  
Joint Clearances ◽  
Constraint Modeling

This paper develops a geometric method to estimate the error space of 3-DOF planar mechanisms with the Minimum Volume Ellipsoid Enclosing (MVEE) approach. Both the joint clearances and actuator errors are considered in this method. Three typical planar parallel mechanisms are used to demonstrate. Error spaces of their serial limbs are analyzed. Thereafter, limb-error-space-constrained mobility of the manipulator, namely, the manipulator error space is analyzed. The MVEE method has been applied to simplify the constraint modeling. A closed-form expression for the manipulator error space is derived. The volume of the manipulator error space is numerically estimated. The approach in this paper is to develop a geometric error analysis method of parallel mechanisms with clear algebraic expressions. Moreover, no forward kinematics computations have been performed in the proposed method, in contrast to the widely used interval analysis method. Although the estimated error space is larger than the actual one, because the enclosing ellipses enlarge the regions of limb error space, the method has an attractive advantage of high computational efficiency.

Mobility Analysis of a Novel 3-5R Parallel Mechanism Family

2004 ◽  
Vol 126 (1) ◽  
pp. 79-82 ◽  
Author(s):  
Q. C. Li ◽  
Z. Huang
Keyword(s):  
Parallel Mechanism ◽  
Degrees Of Freedom ◽  
Screw Theory ◽  
Kinematic Chain ◽  
Parallel Mechanisms ◽  
Mobility Analysis

Mobility analysis of a novel 3-5R parallel mechanism family whose limb consists of a 2R and a 3R parallel subchain is performed by the aid of screw theory. A mobility criterion applicable to such 3-leg parallel mechanisms in which each kinematic chain contains five kinematic pairs is proposed. It is shown that under different structural conditions, the 3-5R parallel mechanism can have 3, 4, or 5 DOF (degrees of freedom). The structural conditions that guarantee the full-cycle mobility are analyzed. The analysis and the method presented in this paper will be helpful in using such a 3-5R parallel mechanism family and introduce new insights into the mobility analysis of parallel mechanisms.

scholarly journals NDF of the near-field of a strip source in orthogonal directions

2020 ◽  
Author(s):  
Rocco Pierri ◽  
Raffaele Moretta
Keyword(s):  
Inverse Problems ◽  
Closed Form ◽  
Degrees Of Freedom ◽  
Near Field ◽  
Closed Form Expression ◽  
Main Difficulty ◽  
Form Expression ◽  
Independent Functions ◽  
Near Zone

<div><div>In the manuscript, we address the problem of evaluating the</div><div>number of degrees of freedom (NDF) of the field radiated by a strip source along all the directions orthogonal to it. </div><div>The NDF represents at the same time the number of independent functions required to represent the data with a given degree of accuracy, and the dimension of the unknowns subspace that can be stably reconstructed. For such reason, the knowledge of the NDF gives insight on the forward and on the inverse problems and it represents one of the metrics to evaluate the achievable performance in the inversion.</div></div><div>The main difficulty arises since in near-zone the eigenvalue</div><div>problem that must be solved for the computation of the NDF,</div><div>involves a non-convolution and non-bandlimited kernel. In the paper, we show how to overcome this drawback and how to obtain a closed-form expression of the NDF which highlights the role played by the configuration parameters.</div>

scholarly journals A general method for rotational averages

2021 ◽  
Vol 2090 (1) ◽  
pp. 012041
Author(s):  
Reed Nessler ◽  
Tuguldur Kh. Begzjav
Keyword(s):  
Closed Form ◽  
Direction Cosine ◽  
Nonlinear Spectroscopy ◽  
Tensor Rank ◽  
Closed Form Expression ◽  
Form Expression ◽  
Oriented Molecules ◽  
Rotationally Invariant ◽  
Arbitrary Tensor ◽  
General Method

Abstract The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over random rotation. We solve this problem for arbitrary tensor rank by deriving a closed-form expression for the rotationally invariant tensor of averaged direction cosine products. From it, we obtain some useful new facts about this tensor. Our results serve to speed the inherently lengthy calculations of nonlinear optics.

Vibration and Sensitivity Analysis of a Beam With a Lumped Mass of Translational and Rotary Inertias

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
D. Wang
Keyword(s):  
Natural Frequency ◽  
Degrees Of Freedom ◽  
Free Vibration Analysis ◽  
Rotary Inertia ◽  
Closed Form Expression ◽  
Lumped Mass ◽  
Form Expression ◽  
Element Analysis ◽  
Discrete Method ◽  
The Finite Element Analysis

The free vibration analysis of a uniform beam carrying a lumped mass with the inclusion of both translational and rotary inertias are performed, and a closed-form expression of the frequency sensitivity with respect to the attachment location of the lumped mass is formulated using the discrete method upon the finite element analysis. By virtually introducing additional degrees of freedom at the mass-attached point, the first-order derivative of the natural frequency can be determined straightforwardly. Comparisons of numerical results from two typical examples show that the rotary inertia of a lumped mass may impose important effects on the natural frequency and its sensitivity. Neglecting the rotary inertia may lead to inaccurate or even erroneous solutions of the beam’s dynamics.

Screw System Analysis of Parallel Mechanisms and Applications to Constraint and Mobility Study

2004 ◽  
Author(s):  
Jian S. Dai ◽  
Zhen Huang ◽  
Harvey Lipkin
Keyword(s):  
Degrees Of Freedom ◽  
System Analysis ◽  
General Mechanism ◽  
Parallel Mechanisms ◽  
Mobility Analysis ◽  
New Approach ◽  
Redundant Constraints ◽  
Overconstrained Mechanisms ◽  
Mobility Study ◽  
System Characteristics

The Kutzbach-Gru¨bler mobility criterion calculates the degrees-of-freedom of a general mechanism. However, the criterion can break down for mechanisms with special geometries, and in particular, the class of so-called overconstrained mechanisms. The problem is that the criterion treats all constraints as active, even redundant constraints, which do not affect the mechanism degrees-of-freedom. This paper examines the screw systems of a parallel mechanism to identify the redundant constraints. The screw system characteristics and relationships are then investigated for physical properties. Then a new approach to mobility analysis is proposed based on screw system decompositions. A new version of the mobility criterion is presented to eliminate the redundant constraints and correctly predict the platform degrees-of-freedom. Several examples of overconstrained mechanisms from the literature illustrate the results.

Mobility of Overconstrained Parallel Mechanisms

2004 ◽  
Vol 128 (1) ◽  
pp. 220-229 ◽  
Author(s):  
Jian S. Dai ◽  
Zhen Huang ◽  
Harvey Lipkin
Keyword(s):  
Physical Properties ◽  
Parallel Mechanism ◽  
Degrees Of Freedom ◽  
General Mechanism ◽  
Parallel Mechanisms ◽  
Mobility Analysis ◽  
New Approach ◽  
Redundant Constraints ◽  
Overconstrained Mechanisms ◽  
System Characteristics

The Kutzbach–Grübler mobility criterion calculates the degrees of freedom of a general mechanism. However, the criterion can break down for mechanisms with special geometries, and in particular, the class of so-called overconstrained parallel mechanisms. The problem is that the criterion treats all constraints as active, even redundant constraints, which do not affect the mechanism degrees of freedom. In this paper we reveal a number of screw systems of a parallel mechanism, explore their inter-relationship and develop an original theoretical framework to relate these screw systems to motion and constraints of a parallel mechanism to identify the platform constraints, mechanism constraints and redundant constraints. The screw system characteristics and relationships are investigated for physical properties and a new approach to mobility analysis is proposed based on decompositions of motion and constraint screw systems. New versions of the mobility criterion are thus presented to eliminate the redundant constraints and accurately predict the platform degrees of freedom. Several examples of overconstrained mechanisms from the literature illustrate the results.

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