Symmetry and invariants of kinematic chains and parallel manipulators

Robotica ◽  
2012 ◽  
Vol 31 (1) ◽  
pp. 61-70 ◽  
Author(s):  
Roberto Simoni ◽  
Celso Melchiades Doria ◽  
Daniel Martins
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Kinematic Analysis ◽  
Group Structure ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Formal Definition ◽  
Kinematic Chains ◽  
Number Synthesis ◽  
Definition Of

SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis of mechanisms and parallel manipulators in early stages of mechanisms design.

The differential calculus of screws: Theory, geometrical interpretation, and applications

2009 ◽  
Vol 223 (6) ◽  
pp. 1449-1468 ◽  
Author(s):  
J J Cervantes-Sánchez ◽  
J M Rico-Martínez ◽  
G González-Montiel ◽  
E J González-Galván
Keyword(s):  
Differential Calculus ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Higher Order ◽  
Abstract Concepts ◽  
Serial Manipulators ◽  
Kinematic Chains ◽  
Finite Displacements ◽  
Typical Shape ◽  
Derivatives Of

This article presents a novel and original formula for the higher-order time derivatives, and also for the partial derivatives of screws, which are successively computed in terms of Lie products, thus leading to the automation of the differentiation process. Through the process and, due to the pure geometric nature of the derivation approach, an enlightening physical interpretation of several screw derivatives is accomplished. Important applications for the proposed formula include higher-order kinematic analysis of open and closed kinematic chains and also the kinematic synthesis of serial and parallel manipulators. More specifically, the existence of a natural relationship is shown between the differential calculus of screws and the Lie subalgebras associated with the expected finite displacements of the end effector of an open kinematic chain. In this regard, a simple and comprehensible methodology is obtained, which considerably reduces the abstraction level frequently required when one resorts to more abstract concepts, such as Lie groups or Lie subalgebras; thus keeping the required mathematical background to the extent that is strictly necessary for kinematic purposes. Furthermore, by following the approach proposed in this article, the elements of Lie subalgebra arise in a natural way — due to the corresponding changes in screws through time — and they also have the typical shape of the so-called ordered Lie products that characterize those screws that are compatible with the feasible joint displacements of an arbitrary serial manipulator. Finally, several application examples — involving typical, serial manipulators — are presented in order to prove the feasibility and validity of the proposed method.

An Approach for Acceleration Analysis of Lower Mobility Parallel Manipulators

2011 ◽  
Vol 3 (1) ◽  
Author(s):  
Haitao Liu ◽  
Tian Huang ◽  
Derek G. Chetwynd
Keyword(s):  
Hessian Matrix ◽  
Parallel Manipulators ◽  
Unified Framework ◽  
Generalized Jacobian ◽  
New Approach ◽  
Kinematic Chains ◽  
Lower Mobility ◽  
Acceleration Analysis ◽  
Definition Of ◽  
Closure Constraints

This paper presents a new approach to the velocity and acceleration analyses of lower mobility parallel manipulators. Building on the definition of the “acceleration motor,” the forward and inverse velocity and acceleration equations are formulated such that the relevant analyses can be integrated under a unified framework that is based on the generalized Jacobian. A new Hessian matrix of serial kinematic chains (or limbs) is developed in an explicit and compact form using Lie brackets. This idea is then extended to cover parallel manipulators by considering the loop closure constraints. A 3-PRS parallel manipulator with coupled translational and rotational motion capabilities is analyzed to illustrate the generality and effectiveness of this approach.

Dexterous Workspace of n-PRRR Planar Parallel Manipulators

2012 ◽  
Vol 4 (3) ◽  
Author(s):  
André Gallant ◽  
Roger Boudreau ◽  
Marise Gallant
Keyword(s):  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Divergence Theorem ◽  
End Effector ◽  
Kinematic Chains ◽  
Prismatic Joint ◽  
Planar Surfaces ◽  
Revolute Joints ◽  
Gauss Divergence Theorem ◽  
Planar Parallel Manipulators

In this work, a method is presented to geometrically determine the dexterous workspace boundary of kinematically redundant n-PRRR (n-PRRR indicates that the manipulator consists of n serial kinematic chains that connect the base to the end-effector. Each chain is composed of two actuated (therefore underlined) joints and two passive revolute joints. P indicates a prismatic joint while R indicates a revolute joint.) planar parallel manipulators. The dexterous workspace of each nonredundant RRR kinematic chain is first determined using a four-bar mechanism analogy. The effect of the prismatic actuator is then considered to yield the workspace of each PRRR kinematic chain. The intersection of the dexterous workspaces of all the kinematic chains is then obtained to determine the dexterous workspace of the planar n-PRRR manipulator. The Gauss divergence theorem applied to planar surfaces is implemented to compute the total dexterous workspace area. Finally, two examples are shown to demonstrate applications of the method.

Innovative Design Based on Prototypes of the Main Mechanism of Table Lamp

2012 ◽  
Vol 198-199 ◽  
pp. 189-192
Author(s):  
Wan Chun Zhou ◽  
Jian Jun Lu
Keyword(s):  
Topological Structure ◽  
Kinematic Chain ◽  
Main Mechanism ◽  
Schematic Diagram ◽  
Design Constraints ◽  
Innovative Design ◽  
Kinematic Chains ◽  
Structure Characteristics ◽  
Practical Design ◽  
Number Synthesis

On the basis of analyzing the topological structure of an original mechanism, it leads to the generalized kinematic chain meeting the same topological structure characteristics of the original mechanism in accordance with the generalized principle of mechanism, number synthesis of kinematic chain and compound hings. Allocation of ground links and performing links in the kinematic chain can get all feasible specialized kinematic chains while considering practical design constraints. In the final step, concrete schematic diagram of mechanism being restored from specialized kinematic chains leads to a series of new mechanism different from the original one. Designers ascertain appropriate new mechanism after selection and analysis.

Enumeration of parallel manipulators

Robotica ◽  
2009 ◽  
Vol 27 (4) ◽  
pp. 589-597 ◽  
Author(s):  
Roberto Simoni ◽  
Andrea Piga Carboni ◽  
Daniel Martins
Keyword(s):  
Group Theory ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Internal Symmetry ◽  
New Method ◽  
Chain Mechanism ◽  
End Effector ◽  
Group Of Automorphisms ◽  
Vertex Graph

SUMMARYIn this paper, we present a new method of enumeration of parallel manipulators with one end-effector. The method consists of enumerating all the manipulators possible with one end-effector that a single kinematic chain can originate. A very useful simplification for kinematic chain, mechanism and manipulator enumeration is their representation through graphs. The method is based on group theory where abstract structures are used to capture the internal symmetry of a structure in the form of automorphisms of a group. The concept used is orbits of the group of automorphisms of a colored vertex graph. The theory and some examples are presented to illustrate the method.

The SOCs Modular Method for Kinematic Analysis of Complicated Parallel Manipulators

2011 ◽  
Vol 52-54 ◽  
pp. 834-841
Author(s):  
Zhi Xin Shi ◽  
Mei Yan Ye ◽  
Yu Feng Luo ◽  
Ting Li Yang
Keyword(s):  
Kinematic Analysis ◽  
Parallel Manipulators ◽  
Initial Guess ◽  
Modular Approach ◽  
Compatibility Conditions ◽  
Kinematic Chains ◽  
Procedural Approach ◽  
Modular Method ◽  
Parallel Kinematic

This paper presents a simple and systematic modular approach for kinematic analysis of complicated Parallel Kinematic Manipulators (in short, PKMs) which coupled degrees are more than 2. (1) Single open chains (in short, SOCs) may be regarded as the basic modules of a PKM. Any PKM can always be decomposed automatically into a set of ordered SOCs, and these SOCs can also be used to recognize the basic kinematic chains contained in it. (2) The kinematic analysis algorithms and the compatibility conditions of the SOC modules are offered. (3) Directly applying the above SOC kinematic modules, the kinematic equations of a PKM can be automatically established. (4) In order to solve kinematic equations of complicated parallel manipulators which coupled degrees are more than 2, a new searching algorithm which requires no initial guess has been presented. The procedural approach is demonstrated in parallel manipulators.

Design and Singularity Analysis of a 3-Translational-DOF In-Parallel Manipulator*

2002 ◽  
Vol 124 (3) ◽  
pp. 419-426 ◽  
Author(s):  
L. Romdhane ◽  
Z. Affi ◽  
M. Fayet
Keyword(s):  
Kinematic Analysis ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Orientation Error ◽  
Kinematic Chains ◽  
Prismatic Joints ◽  
Forward Kinematic ◽  
3D Solid ◽  
Novel Design

In this work, we shall present a novel design of a 3-translational-DOF in-parallel manipulator having 3 linear actuators. Three variable length legs constitute the actuators of this manipulator, whereas two other kinematic chains with passive joints are used to eliminate the three rotations of the platform with respect to the base. This design presents several advantages compared to other designs of similar 3-translational-dof parallel manipulators. First, the proposed design uses only revolute or spherical joints as passive joints and hence, it avoids problems that are inherent to the nature of prismatic joints when loaded in arbitrary way. Second, the actuators are chosen to be linear and to be located in the three legs since this design presents higher rigidity than other. In the second part of this paper, we addressed the problem of kinematic analysis of the proposed in-parallel manipulator. A mixed geometric and vector formulation is used to show that two solutions exist for the forward kinematic analysis. The problem of singularities is also investigated using the same method. In this work, we investigated the singularities of the active legs and the two types of singularity were identified: architectural singularities and configurational singularities. The singularity of the passive chains, used to restrict the motion of the platform to only three translations, is also investigated. In the last part of this paper, we built a 3D solid model of the platform and the amplitude of rotation due to the deformation of the different links under some realistic load was determined. This allowed us to estimate the “orientation error” of the platform due to external moments. Moreover, this analysis allowed us to compare the proposed design (over constrained) with a modified one (not over constrained). This comparison confirmed the conclusion that the over constraint design has a better rigidity.

Type Synthesis of 3-DOF Translational Parallel Manipulators Based on Screw Theory

2004 ◽  
Vol 126 (1) ◽  
pp. 83-92 ◽  
Author(s):  
Xianwen Kong ◽  
Cle´ment M. Gosselin
Keyword(s):  
Screw Theory ◽  
General Procedure ◽  
Linear Equations ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Type Synthesis ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Small Joint ◽  
Actuated Joints

A method is proposed for the type synthesis of 3-DOF (degree-of-freedom) translational parallel manipulators (TPMs) based on screw theory. The wrench systems of a translational parallel kinematic chain (TPKC) and its legs are first analyzed. A general procedure is then proposed for the type synthesis of TPMs. The type synthesis of legs for TPKCs, the type synthesis of TPKCs as well as the selection of actuated joints of TPMs are dealt with in sequence. An approach to derive the full-cycle mobility conditions for legs for TPKCs is proposed based on screw theory and the displacement analysis of serial kinematic chains undergoing small joint motions. In addition to the TPKCs proposed in the literature, TPKCs with inactive joints are synthesized. The phenomenon of dependent joint groups in a TPKC is revealed systematically. The validity condition of actuated joints of TPMs is also proposed. Finally, linear TPMs, which are TPMs whose forward displacement analysis can be performed by solving a set of linear equations, are also revealed.

Kinematic Analysis and Design of Kinematically Redundant Parallel Mechanisms

2004 ◽  
Vol 126 (1) ◽  
pp. 109-118 ◽  
Author(s):  
Jing Wang ◽  
Cle´ment M. Gosselin
Keyword(s):  
Kinematic Analysis ◽  
Main Idea ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Degree Of Freedom ◽  
Redundant Manipulators ◽  
Parallel Mechanisms ◽  
Jacobian Matrices ◽  
Kinematic Chains ◽  
Analysis And Design

This paper addresses the singularity analysis and the design of three new types of kinematically redundant parallel mechanisms, i.e., the four-degree-of-freedom planar and spherical parallel mechanisms and the seven-degree-of-freedom spatial Stewart platform. The main idea in the design of these parallel manipulators is the addition of one redundant degree of freedom in one of the kinematic chains of the nonredundant manipulator. Such manipulators can be used to avoid the singularities inside the workspace of nonredundant manipulators. After describing the geometry of the manipulators, the velocity equations are derived and the expressions for the Jacobian matrices are obtained. Then, the singularity conditions are discussed. Finally, the expressions of the singularity loci of the kinematically redundant mechanisms are obtained and the singularity loci of the nonredundant and redundant manipulators are compared. It is shown here that the conditions for the singularity of the redundant manipulators are reduced drastically relative to the nonredundant ones. As a result, the proposed kinematically redundant parallel manipulators may be of great interest in several applications.

scholarly journals Acceleration Analysis of 3DOF Parallel Manipulators

2013 ◽  
pp. 31-41
Author(s):  
Hassen Nigatu ◽  
Ajit Pal Singh ◽  
Solomon Seid
Keyword(s):  
Hessian Matrix ◽  
Parallel Manipulators ◽  
Unified Framework ◽  
Generalized Jacobian ◽  
New Approach ◽  
Kinematic Chains ◽  
Acceleration Analysis ◽  
Definition Of ◽  
Closure Constraints ◽  
Inverse Velocity

This paper presents a new approach to the velocity and acceleration analyses 3DOF parallel manipulators. Building on the definition of the ‘acceleration motor’, the forward and inverse velocity and acceleration equations are formulated such that the relevant analysis can be integrated under a unified framework that is based on the generalized Jacobian. A new Hessian matrix of serial kinematic chains (or limbs) is developed in an explicit and compact form using Lie brackets. This idea is then extended to cover parallel manipulators by considering the loop closure constraints. A 3- PRS parallel manipulator with coupled translational and rotational motion capabilities is analyzed to illustrate the generality and effectiveness of this approach.

Sign in / Sign up

Export Citation Format

Share Document