Characteristic tetrahedron of wrench singularities for parallel manipulators with three legs

2002 ◽  
Vol 216 (1) ◽  
pp. 81-93 ◽  
Author(s):  
I Ebert-Uphoff ◽  
J-K Lee ◽  
H Lipkin
Keyword(s):  
Parallel Manipulators ◽  
Stewart Platform ◽  
Geometric Interpretation ◽  
Mobile Platform ◽  
Line Geometry ◽  
Important Theorem ◽  
Parallel Platform

A new analysis of wrench singularities is presented for spatial parallel platform manipulators consisting of three legs, with up to two actuators each, and connected to the mobile platform by spherical joints. The analysis also applies to some related manipulators with six legs, such as the 6-3 Gough-Stewart platform. The characteristic tetrahedron is introduced to identify wrench singularities, i.e. configurations where the platform can move infinitesimally with all actuators locked. An important theorem is presented that provides a geometric interpretation of wrench singularities: a manipulator is at a wrench singularity if and only if the characteristic tetrahedron is singular. All cases in which the tetrahedron becomes singular are enumerated, which leads to a classification of wrench singularities. This method is easy to visualize and presents an alternative to standard approaches using line geometry.

Embeddings of locally compact abelian p-groups in Hawaiian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanga Bavuma ◽  
Francesco G. Russo
Keyword(s):  
Algebraic Topology ◽  
Geometric Interpretation ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hawaiian Earring ◽  
Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

SATANIC AND THELEMIC CIRCLES ON KNOTS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350017 ◽  
Author(s):  
G. FLOWERS
Keyword(s):  
Geometric Interpretation ◽  
Second Order ◽  
Geometric Properties ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

A Family of 3-DOF Translational Parallel Manipulators1

2003 ◽  
Vol 125 (2) ◽  
pp. 302-307 ◽  
Author(s):  
Marco Carricato ◽  
Vincenzo Parenti-Castelli
Keyword(s):  
Degrees Of Freedom ◽  
Translational Motion ◽  
Parallel Manipulators ◽  
Geometric Interpretation ◽  
The Other ◽  
Six Degrees Of Freedom ◽  
Revolute Joints ◽  
Rotational Degrees Of Freedom

This article addresses parallel manipulators with fewer than six degrees of freedom, whose use may prove valuable in those applications in which a higher mobility is uncalled for. In particular, a family of 3-dof manipulators containing only revolute joints or at the most revolute and prismatic ones is studied. Design and assembly conditions sufficient to provide the travelling platform with a pure translational motion are determined and two sub-families that fulfill the imposed constraint are found: one is already known in the literature, while the other is original. The new architecture does not exhibit rotation singularities, i.e., configurations in which the platform gains rotational degrees of freedom. A geometric interpretation of the translation singularities is provided.

An Algebraic Method for Direct Position Analysis of the General Stewart Platform Manipulator Robot

2009 ◽  
Vol 626-627 ◽  
pp. 405-410
Author(s):  
Xi Guang Huang ◽  
Guang Pin He ◽  
Q.Z. Liao
Keyword(s):  
Parallel Manipulator ◽  
Algebraic Method ◽  
Stewart Platform ◽  
Mobile Platform ◽  
Analysis Problem ◽  
Position Analysis ◽  
General Geometry ◽  
Six Degree Of Freedom ◽  
Manipulator Robot ◽  
Base Platform

Stewart platform manipulator robot is a six degree of freedom, parallel manipulator, which consists of a base platform, a mobile platform and six limbs connected at six distinct points on the base platform and the mobile platform respectively. The direct position analysis problem of Stewart platform relates to the determination of the mobile platform pose for a given set of the lengths of the limbs. In this paper, we present a concise algebraic method for solving the direct position analysis problem for the fully parallel manipulator with general geometry, often referred to as General Stewart platform manipulator. Based on the presented algebraic method, we derive a 40th degree univariate polynomial from a determinant of 20×20 Sylvester’s matrix, which is relatively small in size. We also obtain a complete set of 40 solutions to the most general Stewart platform. The proposed method is comparatively concise and reduces the computational burden. Finally the method is demonstrated by a numerical example.

Analysis of Varying Natural Frequencies and Damping Ratios of a Sample Parallel Manipulator Throughout its Workspace Using Linearized Equations of Motion

2001 ◽  
Author(s):  
Kris Kozak ◽  
Imme Ebert-Uphoff ◽  
William Singhose
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Parallel Architecture ◽  
Robotic Manipulators ◽  
Parallel Manipulators ◽  
Dynamic Equations ◽  
Extreme Variation

Abstract This article investigates the dynamic properties of robotic manipulators of parallel architecture. In particular, the dependency of the dynamic equations on the manipulator’s configuration within the workspace is analyzed. The proposed approach is to linearize the dynamic equations locally throughout the workspace and to plot the corresponding natural frequencies and damping ratios. While the results are only applicable for small velocities of the manipulator, they present a first step towards the classification of the nonlinear dynamics of parallel manipulators. The method is applied to a sample manipulator with two degrees-of-freedom. The corresponding numerical results demonstrate the extreme variation of its natural frequencies and damping ratios throughout the workspace.

Determination of Constant Orientation Workspace of a Stewart Platform by Geometrical Method

2015 ◽  
Vol 813-814 ◽  
pp. 997-1001 ◽  
Author(s):  
S. Gokul Narasimhan ◽  
R. Shrivatsan ◽  
K. Venkatasubramanian ◽  
Anjan Kumar Dash
Keyword(s):  
Algebraic Method ◽  
Parallel Manipulators ◽  
Stewart Platform ◽  
Closed Form Expression ◽  
Robot Design ◽  
Form Expression ◽  
Novel Method ◽  
Workspace Optimization ◽  
Orientation Workspace

Determination of workspace is one of the main considerations in the design of any robot since the workspace geometry is considered a fundamental issue for robot design. This also plays a crucial role in trajectory planning. Among parallel manipulators, 6-DOF Stewart platforms is the most researched and widely used robot. However, till date there is no closed form expression of workspace volume for Stewart platform. In this paper, a novel method is proposed to find out the workspace volume of Stewart platform. In this paper, individual workspace of each leg of the manipulator (P-U-S) is determined and then translated by a common distance towards their geometrical center thus generating constant orientation workspace. To determine the workspace volume, geometric intersection of the six spheres is computed. This results in workspace of definite shape and size, whose volume is calculated using simple formulae. It is observed that the geometric way of determination of workspace area is computationally less tedious than the algebraic method. This also helps a lot for workspace optimization of such manipulators.

Workspace formulation of planar wire-actuated parallel manipulators

Robotica ◽  
2010 ◽  
Vol 29 (4) ◽  
pp. 607-617 ◽  
Author(s):  
Derek McColl ◽  
Leila Notash
Keyword(s):  
External Force ◽  
Parallel Manipulators ◽  
Applied Force ◽  
Mobile Platform ◽  
Static Force ◽  
Force Analysis ◽  
Wide Range

SUMMARYIn this paper, a generalized form of the antipodal method from multi-finger grasping is presented and implemented for investigating the workspace of a wide range of planar wire-actuated parallel manipulators. Manipulators with distinct wire attachment points on the base and mobile platform are considered, in the absence and presence of external force. The examined workspaces are verified with the corresponding workspaces developed using static force analysis. By applying an external force, modeled as a wire for the antipodal method, the characteristics of the manipulator could be altered by enlarging its workspace in the direction of the applied force.

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