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

Journal of Mechanisms and Robotics ◽

10.1115/1.4051060 ◽

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.

Download Full-text

Analysis of Force Distribution in Grasps Using Augmentation

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1996_210_165_02 ◽

1996 ◽

Vol 210 (1) ◽

pp. 15-22 ◽

Cited By ~ 15

Author(s):

J S Dai ◽

D R Kerr

Keyword(s):

Rigid Body ◽

Equilibrium Condition ◽

Jacobian Matrix ◽

Point Contact ◽

Force Distribution ◽

Mathematical Basis ◽

New Approach ◽

Contact Points ◽

Geometric Compatibility ◽

Force Equilibrium

A new approach to the analysis of statically indeterminate restraint of a rigid body with any arrangement of point contact is presented in this paper. The paper associates the elasticity at restraint contacts with geometric compatibility of the contact points and constructs elastic compatibility equations, which are complementary to the restraint equations. The equations so obtained are then used to augment the restraint equations and lead to an agumented Jacobian matrix. The new approach enables grasps to be analysed and synthesized in a constraint of combined elasticity and geometric compatibility, in addition to the force equilibrium condition. This gives a mathematical basis for the analysis of force distribution of the statically indeterminate restraint. Detailed reasoning and derivations are given followed by both planar and spatial examples.

Download Full-text

Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures

Journal of Mechanisms and Robotics ◽

10.1115/1.4005336 ◽

2012 ◽

Vol 4 (1) ◽

Cited By ~ 40

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.

Download Full-text

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

Volume 6: 35th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2011-48226 ◽

2011 ◽

Cited By ~ 2

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.

Download Full-text

On the Design of Singularity-Free Cable-Driven Parallel Mechanism Based on Grassmann Geometry

Volume 4: 36th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2012-71076 ◽

2012 ◽

Author(s):

Yu Zou ◽

Yuru Zhang ◽

Yaojun Zhang

Keyword(s):

Parallel Mechanism ◽

Jacobian Matrix ◽

Singularity Analysis ◽

Geometric Meaning ◽

Parallel Mechanisms ◽

Line Geometry ◽

Singular Configurations ◽

Negative Effect ◽

Grassmann Line Geometry ◽

Grassmann Geometry

This paper deals with the design of singularity-free cable-driven parallel mechanism. Due to the negative effect on the performance, singularities should be avoided in the design. The singular configurations of mechanisms can be numerically determined by calculating the rank of its Jacobian matrix. However, this method is inefficient and non-intuitive. In this paper, we investigate the singularities of planar and spatial cable-driven parallel mechanisms using Grassmann line geometry. Considering cables as line vectors in projective space, the singularity conditions are identified with clear geometric meaning which results in useful method for singularity analysis of the cable-driven parallel mechanisms. The method is applied to 3-DOF planar and 6-DOF spatial cable-driven mechanisms to determine their singular configurations. The results show that the singularities of both mechanisms can be eliminated by changing the dimensions of the mechanisms or adding extra cables.

Download Full-text

Analysis of Singularities of a 3DOF Parallel Manipulator Based on a Novel Geometrical Method

Volume 1: Advanced Energy Systems, Advanced Materials, Aerospace, Automation and Robotics, Noise Control and Acoustics, and Systems Engineering ◽

10.1115/esda2006-95592 ◽

2006 ◽

Author(s):

Hodjat Pendar ◽

Hajir Roozbehani ◽

Hoda Sadeghian ◽

Hassan Zohoor

Keyword(s):

Angular Velocity ◽

Parallel Mechanism ◽

Parallel Manipulator ◽

Jacobian Matrix ◽

The Novel ◽

Geometrical Method ◽

Novel Method ◽

Grassmann Geometry ◽

Constraint Plane ◽

Definition Of

In this article singular points of a parallel manipulator are obtained based on a novel geometrical method. Here we introduce the constrained plain method (CPM) and some of its application in parallel mechanism. Given the definition of constraint plane (CP) and infinite constraint plane (ICP) the dependency conditions of constraints is achieved with the use of a new theorem based on the Ceva geometrical theorem. The direction of angular velocity of a body is achieved by having three ICPs with the use of another theorem. Finally, with the use of the above two novel theorems singularities of the 3UPF_PU mechanism are obtained. It should be emphasized that this method is completely geometrical, involving no complex or massive calculations. In the previous methods based on the Grassmann Geometry, the mechanism needs to be statically analyzed at first, so that the Inverse Jacobian Matrix is achieved, and then the Plucker-Vector is derived. This task is somewhat inconvenient and in the end there are plenty of conditions remained to be pondered in order to obtain the singularity conditions, while the novel method introduced here, involves no tiring calculations neither the analysis of numerous conditions and yields the answer quickly.

Download Full-text