Grassmann, geometry and mechanics

2010 ◽  
pp. 287-302
Author(s):  
John Browne
Keyword(s):  
Grassmann Geometry

Singularity Loci, Bifurcated Evolution Routes, and Configuration Transitions of Reconfigurable Legged Mobile Lander from Adjusting, Landing, to Roving

2021 ◽  
pp. 1-11
Author(s):  
Youcheng Han ◽  
Caizhi Zhou ◽  
Weizhong Guo
Keyword(s):  
Rigid Body ◽  
Jacobian Matrix ◽  
Impact Force ◽  
Mutual Authentication ◽  
Selection Principle ◽  
Passive Mechanism ◽  
Grassmann Geometry

Abstract This paper presents the reconfigurable legged mobile lander (ReLML) with its modes from adjusting, landing, to roving. Based on the invented metamorphic variable-axis revolute hinge, the actuated link has three alternative phases of rotating around either of two orthogonal topological axes or locking itself to the base as a rigid body. This property enables the ReLML to switch among three modes and within two driving states (as the adjusting and roving modes are active mechanisms driven by motors, while the landing truss is regarded as a passive mechanism driven by the touchdown impact force exerted on footpad). The unified differential kinematics for the ReLML is established by the screw-based Jacobian modeling, unifying both active and passive operation phases throughout all modes. Afterward, the distributions of workspaces and singularity loci in three modes are discussed for the multi-solution sake, and the selection principle of the practicable solution pattern is proposed to obtain the actual workspace, singularity loci, and configurations. The results stemming from the Jacobian-matrix-based method and the Grassmann-geometry-based method give mutual authentication and match well. Finally, as prospects for promising applications, four bifurcated evolution routes and configuration transitions are figured out and compared.

Grassmann geometry on symmetric spaces

1999 ◽  
Vol 44 (1) ◽  
pp. 41-44
Author(s):  
Quanqin Jin
Keyword(s):  
Symmetric Spaces ◽  
Grassmann Geometry

scholarly journals Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures

2012 ◽  
Vol 4 (1) ◽  
Author(s):  
Semaan Amine ◽  
Mehdi Tale Masouleh ◽  
Stéphane Caro ◽  
Philippe Wenger ◽  
Clément Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Type Synthesis ◽  
Constraint Analysis ◽  
Fixed Direction ◽  
Geometric Singularity ◽  
Grassmann Geometry ◽  
Grassmann Cayley Algebra

This paper deals with the singularity analysis of parallel manipulators with identical limb structures performing Schönflies motions, namely, three independent translations and one rotation about an axis of fixed direction (3T1R). Eleven architectures obtained from a recent type synthesis of such manipulators are analyzed. The constraint analysis shows that these architectures are all overconstrained and share some common properties between the actuation and the constraint wrenches. The singularities of such manipulators are examined through the singularity analysis of the 4-RUU parallel manipulator. A wrench graph representing the constraint wrenches and the actuation forces of the manipulator is introduced to formulate its superbracket. Grassmann–Cayley Algebra is used to obtain geometric singularity conditions. Based on the concept of wrench graph, Grassmann geometry is used to show the rank deficiency of the Jacobian matrix for the singularity conditions. Finally, this paper shows the general aspect of the obtained singularity conditions and their validity for 3T1R parallel manipulators with identical limb structures.

Singularity Analysis of the 4-RUU Parallel Manipulator Based on Grassmann-Cayley Algebra and Grassmann Geometry

2011 ◽  
Author(s):  
Semaan Amine ◽  
Mehdi Tale Masouleh ◽  
Ste´phane Caro ◽  
Philippe Wenger ◽  
Cle´ment Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Graph Representation ◽  
Fixed Direction ◽  
Cayley Algebra ◽  
Geometric Singularity ◽  
Grassmann Geometry ◽  
Grassmann Cayley Algebra

This paper deals with the singularity analysis of parallel manipulators with identical limb structures performing Scho¨nflies motions, namely, three independent translations and one rotation about an axis of fixed direction. The study is developed through the singularity analysis of the 4-RUU parallel manipulator. The 6 × 6 Jacobian matrix of such manipulators contains two lines at infinity, namely, two constraint moments, among its six Plu¨cker lines. The Grassmann-Cayley Algebra is used to obtain geometric singularity conditions. However, due to the presence of lines at infinity, the rank deficiency of the Jacobian matrix for the singularity conditions is not easy to grasp. Therefore, a wrench graph representation for some singularity conditions emphasizes the linear dependence of the Plu¨cker lines of the Jacobian matrix and highlights the correspondence between Grassmann-Cayley algebra and Grassmann geometry.

scholarly journals Conceptual Design of Schoenflies Motion Generators Based on the Wrench Graph

2013 ◽  
Author(s):  
Semaan Amine ◽  
Latifah Nurahmi ◽  
Philippe Wenger ◽  
Stéphane Caro
Keyword(s):  
Conceptual Design ◽  
Parallel Manipulator ◽  
Screw Theory ◽  
Parallel Manipulators ◽  
Design Stage ◽  
The Subject ◽  
Grassmann Geometry ◽  
Kinematic Singularities ◽  
General Configuration

The subject of this paper is about the conceptual design of parallel Schoenflies motion generators based on the wrench graph. By using screw theory and Grassmann geometry, some conditions on both the constraint and the actuation wrench systems are generated for the assembly of limbs of parallel Schoenflies motion generators, i.e., 3T1R parallel manipulators. Those conditions are somehow related to the kinematic singularities of the manipulators. Indeed, the parallel manipulator should not be in a constraint singularity in the starting configuration for a valid architecture, otherwise it cannot perform the required motion pattern. After satisfying the latter condition, the parallel manipulator should not be in an actuation singularity in a general configuration, otherwise the obtained parallel manipulator is permanently singular. Based on the assembly conditions, six types of wrench graphs are identified and correspond to six typical classes of 3T1R parallel manipulators. The geometric properties of these six classes are highlighted. A simplified expression of the superbracket decomposition is obtained for each class, which allows the determination and the comparison of the singularities of 3T1R parallel manipulators at their conceptual design stage. The methodology also provides new architectures of parallel Schoenflies motion generators based on the classification of wrench graphs and on their singularity conditions.

Grassmann geometry on the 3-dimensional non-unimodular Lie groups

2019 ◽  
Vol 48 (2) ◽  
pp. 385-406
Author(s):  
Jun-ichi INOGUCHI ◽  
Hiroo NAITOH
Keyword(s):  
Lie Groups ◽  
3 Dimensional ◽  
Grassmann Geometry
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