Singularity Equations of Gough-Stewart Platforms Using a Minimal Set of Geometric Parameters

2007 ◽  
Author(s):  
Qimi Jiang ◽  
Cle´ment M. Gosselin
Keyword(s):  
Singularity Analysis ◽  
Geometric Optimization ◽  
Geometric Parameters ◽  
Minimal Set ◽  
New Approach ◽  
Practical Point ◽  
Fixed Frame ◽  
Solid Foundation ◽  
Stewart Platforms ◽  
Mobile Frame

So far, in the derivation of the singularity equations of Gough-Stewart platforms, all research works defined the mobile frame by making its origin coincide with the considered point on the platform. One problem can be that the obtained singularity equation contains too many geometric parameters and is not convenient for singularity analysis, especially not convenient for geometric optimization. Another problem can be that the obtained singularity equation cannot be used directly in practice. To solve these problems, this work presents a new approach to derive the singularity equation of the Gough-Stewart platform. The main point is that the origin of the mobile frame is separated from the considered point and chosen to coincide with a special point of the platform in order to minimize the geometric parameters defining the platform. Similarly, by defining a proper fixed frame, the geometric parameters defining the base can also be minimized. In this way, no matter which practical point of the platform is chosen as the considered point, the obtained singularity equation contains only a minimal set of geometric parameters and becomes a solid foundation for the geometric optimization based on singularity analysis.

Singularity Equations of Gough–Stewart Platforms Using a Minimal Set of Geometric Parameters

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
Qimi Jiang ◽  
Clément M. Gosselin
Keyword(s):  
Singularity Analysis ◽  
Geometric Optimization ◽  
Geometric Parameters ◽  
Minimal Set ◽  
New Approach ◽  
Practical Point ◽  
Fixed Frame ◽  
Solid Foundation ◽  
Stewart Platforms ◽  
Mobile Frame

So far, in the derivation of the singularity equations of Gough–Stewart platforms, all researchers defined the mobile frame by making its origin coincide with the considered point on the platform. One problem can be that the obtained singularity equation contains too many geometric parameters and is not convenient for singularity analysis, especially not convenient for geometric optimization. Another problem can be that the obtained singularity equation cannot be used directly in practice. To solve these problems, this work presents a new approach to derive the singularity equation of the Gough–Stewart platform. The main point is that the origin of the mobile frame is separated from the considered point and chosen to coincide with a special point on the platform in order to minimize the geometric parameters defining the platform. Similarly, by defining a proper fixed frame, the geometric parameters defining the base can also be minimized. In this way, no matter which practical point of the platform is chosen as the considered point, the obtained singularity equation contains only a minimal set of geometric parameters and becomes a solid foundation for the geometric optimization based on singularity analysis.

Singularity analysis of a 3-DOF parallel manipulator with R-P-S joint structure

Robotica ◽  
2005 ◽  
Vol 24 (1) ◽  
pp. 131-142 ◽  
Author(s):  
Alexei Sokolov ◽  
Paul Xirouchakis
Keyword(s):  
Inverse Kinematics ◽  
Parallel Manipulator ◽  
Optimization Procedure ◽  
Singularity Analysis ◽  
Robot Design ◽  
Geometrical Method ◽  
New Approach ◽  
Joint Structure ◽  
Working Space ◽  
Robot Trajectory

This paper presents a singularity analysis for a 3-DOF parallel manipulator with R-P-S (Revolute-Prismatic-Spherical) joint structure. All three types of singularities are investigated with most attention paid for direct kinematics singularities (DKS). The loci of inverse kinematics and combined singularities are identified using a new approach. The equation of DKS is defined first from the condition of existence of an instantaneous motion. The geometrical method is used to find the loci of trajectories corresponding to DKS-s. As a result of these investigations, an optimization procedure was proposed of a robot design in order to have an enlarged singularity free part of the working space. The construction of a singularity free path is discussed without changing the robot trajectory by selecting the appropriate inverse kinematics task solution.

Instantaneous kinematics and singularity analysis of three-legged parallel manipulators

Robotica ◽  
2004 ◽  
Vol 22 (2) ◽  
pp. 189-203 ◽  
Author(s):  
Anjan Kumar Dash ◽  
I-Ming Chen ◽  
Song Huat Yeo ◽  
Guilin Yang
Keyword(s):  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Physical Structure ◽  
Inverse Kinematic ◽  
New Approach ◽  
Kinematics Model ◽  
Passive Joint ◽  
Geometric Conditions ◽  
Joint Plane ◽  
Inverse Kinematic Analysis

Instantaneous kinematics and singularity analysis of a class of three-legged, 6-DOF parallel manipulators are addressed in this paper. A generic method of derivation of reciprocal screw and consequently, the instantaneous kinematics model is presented. The advantage of this formulation is that the instantaneous kinematics model possesses well-defined geometric meaning and algebraic structure. Singularity analysis is performed under three categories, namely forward, inverse and combined singularities. A new concept of Passive Joint Plane is introduced to correlate the physical structure of the manipulator and these geometric conditions. In the inverse kinematic analysis, a new approach is introduced. At each leg end point a characteristic parallel- epiped is defined whose sides are the linear velocity components from three main joints of the leg. An inverse singularity occurs when the volume of this parallelepiped becomes zero. Examples are demonstrated using RRRS and RPRS-type parallel manipulators.

Statics, Instantaneous Kinematics and Singularity Analysis of Planar Parallel Manipulators via Grassmann-Cayley Algebra

2014 ◽  
Vol 532 ◽  
pp. 378-381 ◽  
Author(s):  
Ke Fei Wen ◽  
Jeh Won Lee
Keyword(s):  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
New Approach ◽  
Coordinate Method ◽  
Cayley Algebra ◽  
Symbolic Formula ◽  
Grassmann Cayley Algebra ◽  
Planar Parallel Manipulators

The wrench Jacobian matrix plays an important role in statics and singularity analysis of planar parallel manipulators (PPMs). It is easy to obtain this matrix based on plücker coordinate method. In this paper, a new approach is proposed to the analysis of the forward and inverse wrench Jacobian matrix used by Grassmann-Cayley algebra (GCA). A symbolic formula for the inverse statics and a coordinate free formula for the singularity analysis are obtained based on this Jacobian. As an example, this approach is implemented for the 3-RPR PPMs.

scholarly journals Political Legitimacy as an Existential Predicament

2021 ◽  
pp. 009059172110478
Author(s):  
Thomas Fossen
Keyword(s):  
Political Legitimacy ◽  
Point Of View ◽  
Good Judgment ◽  
New Approach ◽  
Practical Point ◽  
The Face

This essay contributes to developing a new approach to political legitimacy by asking what is involved in judging the legitimacy of a regime from a practical point of view. It is focused on one aspect of this question: the role of identity in such judgment. I examine three ways of understanding the significance of identity for political legitimacy: the foundational, associative, and agonistic picture. Neither view, I claim, persuasively captures the dilemmas of judgment in the face of disagreement and uncertainty about who “I” am and who “we” are. I then propose a composite, pragmatic picture. This view casts the question of political legitimacy as an existential predicament: it is fundamentally a question about who you are—both as a person and as a member of collectives. The pragmatic picture integrates rational, prudential, and ethical qualities of good judgment that were heretofore associated with mutually exclusive ways of theorizing legitimacy. It also implies that the question of legitimacy cannot be resolved philosophically.

Geometrical method to determine the reciprocal screws and applications to parallel manipulators

Robotica ◽  
2009 ◽  
Vol 27 (6) ◽  
pp. 929-940 ◽  
Author(s):  
Jianguo Zhao ◽  
Bing Li ◽  
Xiaojun Yang ◽  
Hongjian Yu
Keyword(s):  
Parallel Manipulator ◽  
Screw Theory ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Algebraic Manipulation ◽  
Robot Kinematics ◽  
Geometrical Approach ◽  
New Approach ◽  
Geometrical Relation ◽  
Manipulator Design

SUMMARYScrew theory has demonstrated its wide applications in robot kinematics and statics. We aim to propose an intuitive geometrical approach to obtain the reciprocal screws for a given screw system. Compared with the traditional Plücker coordinate method, the new approach is free from algebraic manipulation and can be used to obtain the reciprocal screws just by inspecting the structure of manipulator. The approach is based on three observations that describe the geometrical relation for zero pitch screw and infinite pitch screw. Based on the observations, the reciprocal screw systems of several common kinematic elements are analyzed, including usual kinematic pairs and chains. We also demonstrate usefulness of the geometrical approach by a variety of applications in mobility analysis, Jacobian formulation, and singularity analysis for parallel manipulator. This new approach can facilitate the parallel manipulator design process and provide sufficient insights for existing manipulators.

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