Compliance Analysis of Mechanisms with Spatial Continuous Compliance in the Context of Screw Theory and Lie Groups

2010 ◽  
Vol 224 (11) ◽  
pp. 2493-2504 ◽  
Author(s):  
X Ding ◽  
J S Dai
Keyword(s):  
Case Studies ◽  
Lie Groups ◽  
Screw Theory ◽  
Elastic Beam ◽  
Parallel Mechanisms ◽  
Finite Segment ◽  
Numerical Examples ◽  
Spatial Compliance ◽  
Eigenvectors And Eigenvalues

This paper investigates the compliance effect on both serial and parallel mechanisms based on study of deflections of a finite segment of elastic beam with spatial compliance and applies eigenvectors and eigenvalues to identify principal screws in the mechanisms and parallel devices with spatial continuous compliance. With the analysis, compliance characteristics of both serial mechanisms and parallel devices can be identified with effect of compliance. Case studies are presented with numerical examples.

scholarly journals Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

2020 ◽  
Author(s):  
Peng Sun ◽  
Yanbiao Li ◽  
Ke Chen ◽  
Wen-Tao Zhu ◽  
Qi Zhong ◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Screw Theory ◽  
Hessian Matrix ◽  
Parallel Mechanisms ◽  
Analysis Method ◽  
Kinematics Analysis ◽  
Hybrid Mechanisms ◽  
Parallel Configuration

Abstract The advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms, and the generalized analysis method and concise kinematics transfer matrix are obtained. First, according to the kinematics analysis of serial mechanisms, the basic principles of Lie groups Lie algebras in dealing with the spatial switching and differential operations of screw vectors are briefly explained. Then, based on the standard ideas of Lie operations, the method for kinematics analysis of parallel mechanisms is derived, and Jacobian matrix and Hessian matrix are formulated both recursively and in closed form. After that, according to the mapping relationship between the parallel joints and the corresponding equivalent series joints, a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are studied. A case study is performed to verify the calculated matrices, in which a humanoid hybrid robotic arm with the parallel-series-parallel configuration is taken as an example. Simulation experiment results show that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practicable.

scholarly journals Type Synthesis of 2T1R Decoupled Parallel Mechanisms Based on Lie Groups and Screw Theory

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Shuwei Qu ◽  
Ruiqin Li ◽  
Shaoping Bai
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Screw Theory ◽  
New Method ◽  
Parallel Mechanisms ◽  
Type Synthesis ◽  
Compact Structure ◽  
Selection Of

Decoupled parallel mechanisms (DPMs) have the characteristics of compact structure and simple control with wide applications. This paper presents a new method of type synthesis for DPMs by virtue of Lie groups and screw theory. The method consists of synthesis at limb level and configuration level. At limb level, Lie group is used to synthesize the limbs with required DOFs. At configuration level, screw theory is adopted to determine configuration with synthesized limbs that satisfy the type synthesis criteria of DPMs. The type synthesis criteria including limb decoupling and selection of the driving pairs are presented. Upon the formulation, the procedure of type synthesis of DPMs is developed. Type synthesis is conducted with the proposed method, which leads to new spatial and planar fully decoupled 2T1R mechanisms.

On the Maximization of Joint Velocities and Generalized Reactions in the Workspace and Singularity Analysis of Parallel Mechanisms

Robotica ◽  
2018 ◽  
Vol 37 (4) ◽  
pp. 675-690 ◽  
Author(s):  
Pavel Laryushkin ◽  
Victor Glazunov ◽  
Ksenia Erastova
Keyword(s):  
Case Studies ◽  
External Load ◽  
Parallel Mechanism ◽  
Degrees Of Freedom ◽  
Screw Theory ◽  
State Of The Art ◽  
Singularity Analysis ◽  
The State ◽  
Parallel Mechanisms ◽  
Different Types

SummaryAn approach for calculating the maximum possible absolute values of joint velocities or generalized reactions in a leg of a parallel mechanism has been considered in this paper. The Jacobian analysis and the Screw theory-based methods have been used to acquire the result. These values are calculated for the “worst” directions of the external load or end-effector’s velocity for each leg. The feasibility of using these parameters as the measures of closeness to different types of parallel mechanism singularity is discussed. Further, how this approach is related to the state-of-the-art methods has been illustrated. The key aspect of the discussed approach is that the normalization of vectors or screws is carried out separately for angular and linear components. One possible advantage of such an approach is that it deals only with the kinematic and statics of the mechanism while still providing physically meaningful and practically applicable measures. Case studies of a 3-Degrees Of Freedom translational parallel mechanism and a planar parallel mechanism are presented for illustration and comparison.

scholarly journals Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Peng Sun ◽  
Yanbiao Li ◽  
Ke Chen ◽  
Wentao Zhu ◽  
Qi Zhong ◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Screw Theory ◽  
Hessian Matrix ◽  
Parallel Mechanisms ◽  
Analysis Method ◽  
Kinematics Analysis ◽  
Hybrid Mechanisms ◽  
Parallel Configuration

AbstractAdvanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms, and the generalized analysis method and concise kinematics transfer matrix are obtained. In this study, first, according to the kinematics analysis of serial mechanisms, the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors. Then, based on the standard ideas of Lie operations, the method for kinematics analysis of parallel mechanisms is derived, and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form. Then, according to the mapping relationship between the parallel joints and corresponding equivalent series joints, a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined. A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example. The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.

scholarly journals A New Method for Type Synthesis of 2R1T and 2T1R 3-DOF Redundant Actuated Parallel Mechanisms with Closed Loop Units

2020 ◽  
Vol 33 (1) ◽  
Author(s):  
Yongquan Li ◽  
Yang Zhang ◽  
Lijie Zhang
Keyword(s):  
Closed Loop ◽  
Screw Theory ◽  
Line Graph ◽  
Degree Of Freedom ◽  
Parallel Mechanisms ◽  
Type Synthesis ◽  
Allocation Criteria ◽  
Grassmann Line Geometry ◽  
Moving Platform ◽  
Atlas Method

Abstract The current type synthesis of the redundant actuated parallel mechanisms is adding active-actuated kinematic branches on the basis of the traditional parallel mechanisms, or using screw theory to perform multiple getting intersection and union to complete type synthesis. The number of redundant parallel mechanisms obtained by these two methods is limited. In this paper, based on Grassmann line geometry and Atlas method, a novel and effective method for type synthesis of redundant actuated parallel mechanisms (PMs) with closed-loop units is proposed. Firstly, the degree of freedom (DOF) and constraint line graph of the moving platform are determined successively, and redundant lines are added in constraint line graph to obtain the redundant constraint line graph and their equivalent line graph, and a branch constraint allocation scheme is formulated based on the allocation criteria. Secondly, a scheme is selected and redundant lines are added in the branch chains DOF graph to construct the redundant actuated branch chains with closed-loop units. Finally, the branch chains that meet the requirements of branch chains configuration criteria and F&C (degree of freedom & constraint) line graph are assembled. In this paper, two types of 2 rotational and 1 translational (2R1T) redundant actuated parallel mechanisms and one type of 2 translational and 1 rotational (2T1R) redundant actuated parallel mechanisms with few branches and closed-loop units were taken as examples, and 238, 92 and 15 new configurations were synthesized. All the mechanisms contain closed-loop units, and the mechanisms and the actuators both have good symmetry. Therefore, all the mechanisms have excellent comprehensive performance, in which the two rotational DOFs of the moving platform of 2R1T redundant actuated parallel mechanism can be independently controlled. The instantaneous analysis shows that all mechanisms are not instantaneous, which proves the feasibility and practicability of the method.

Determination of Singularity-Free Zones in the Workspace of Planar 3-P̱RR Parallel Mechanisms

2006 ◽  
Vol 129 (6) ◽  
pp. 649-652 ◽  
Author(s):  
Mehdi Tale Masouleh ◽  
Clément Gosselin
Keyword(s):  
Parallel Mechanisms ◽  
Numerical Examples ◽  
Mathematical Derivation

This paper presents an algorithm for the determination of singularity-free zones in the workspace of the planar 3-P̱RR mechanism. The mathematical derivation of the algorithm is first given. Numerical examples are then included to demonstrate the application of the proposed approach.

scholarly journals The Krylov problem in the case of a beam on Vlasov inertial foundation

2018 ◽  
Vol 19 (6) ◽  
pp. 728-736
Author(s):  
Wacław Szcześniak ◽  
Magdalena Ataman
Keyword(s):  
Closed Form ◽  
Compressive Force ◽  
Elastic Beam ◽  
Forced Vibrations ◽  
Critical Force ◽  
Winkler Foundation ◽  
Static Deflection ◽  
Numerical Examples ◽  
Moving Force ◽  
Axial Forces

The paper deals with vibrations of the elastic beam caused by the moving force traveling with uniform speed. The function defining the pure forced vibrations (aperiodic vibrations) is presented in a closed form. Dynamic deflection of the beam caused by moving force is compared with the static deflection of the beam subjected to the force , and compressed by axial forces . Comparing equations (9) and (13), it can be concluded that the effect on the deflection of the speed of the moving force is the same as that of an additional compressive force . Selected problems of stability of the beam on the Winkler foundation and on the Vlasov inertial foundation are discussed. One can see that the critical force of the beam on Vlasov foundation is greater than in the case of Winkler's foundation. Numerical examples are presented in the paper

A Systematical Approach for Structure Synthesis of Parallel Mechanisms Based on Single-Open-Chain Modules and Its Application

2009 ◽  
Author(s):  
Ting-Li Yang ◽  
An-Xin Liu ◽  
Qiong Jin ◽  
Yu-Feng Luo ◽  
Lu-Bin Hang ◽  
...  
Keyword(s):  
Parallel Mechanism ◽  
Systematic Approach ◽  
Screw Theory ◽  
Parallel Mechanisms ◽  
Type Synthesis ◽  
Open Chain ◽  
Structure Synthesis ◽  
Vector Algebra ◽  
Algebra Theory ◽  
Position And Orientation

Based on previous research results presented by authors, this paper proposes a novel systematic approach for structure synthesis of all parallel mechanisms (excluding Bennett mechanism etc), which is totally different from the approaches based on screw theory and based on displacement subgroup. Main characteristics of this approach are: (a) the synthesized mechanisms are non-instantaneous ones, and (b) only simple mathematical tools (vector algebra, theory of sets, etc.) are used. Main steps of this approach include: (1) Determining functional and structural requirements of the parallel mechanism to be synthesized, such as position and orientation characteristic (POC) matrix, degree of freedom (DOF), etc. (2) Type synthesis of branches. (3) Assembling of branches (determining the geometry constraint conditions among the branches attached between the moving platform and the frame, and checking the DOF). (4) Identifying the inactive joints. (5) Selecting the actuating joints. In order to illustrate the whole procedure, the type synthesis of spherical parallel mechanisms is studied using this approach.

REAL-TIME GAIT PLANNING FOR THE HUMANOID ROBOT Rh-1 USING THE LOCAL AXIS GAIT ALGORITHM

2009 ◽  
Vol 06 (01) ◽  
pp. 71-91 ◽  
Author(s):  
MARIO ARBULU ◽  
CARLOS BALAGUER
Keyword(s):  
Real Time ◽  
Lie Groups ◽  
Humanoid Robot ◽  
Screw Theory ◽  
Step Length ◽  
Center Of Gravity ◽  
Joint Torque ◽  
Continuous Change ◽  
Motion Generation ◽  
Table Model

This paper presents the 3D foot and center of gravity motion planning for the humanoid robot called the "local axis gait" (LAG) algorithm. It permits walking on different kinds of surfaces, such as planes, ramps or stairs. Furthermore, continuous change of the step length and orientation in real time will be possible, due to the real-time linear dynamics model of the walking pattern of the humanoid. The robot model is based on the cart table formulation for planning the center of gravity (COG) and zero moment point (ZMP) motion. The proposed algorithm takes into account physical robot constraints such as joint angles, angular velocity and torques. Torques are computed by the Lagrange method under screws and Lie groups. The LAG is divided into several stages: computation of the footprints; the decision of the ZMP limits around the footprints; the dynamic humanoid COG motion generation based on the cart table model; and joining the footprints of the swing foot by splines. In this way it is possible to generate each step online, using the desired footprints as input. In order to compute the joint torque limits, the Lagrangian method is used under the Lie groups and screw theory. The paper presents and discusses some successful results on the LAG in the full-size humanoid robot Rh-1 developed in the Roboticslab of University Carlos III of Madrid.

A New Index of Static Actuation Force Sensitivity of Mechanisms Based on Unified Forward Jacobian Matrix

2020 ◽  
Vol 12 (6) ◽  
Author(s):  
Xu Wang ◽  
Weizhong Guo ◽  
Youcheng Han
Keyword(s):  
Jacobian Matrix ◽  
Screw Theory ◽  
Structural Constraints ◽  
Parallel Mechanisms ◽  
Analysis Method ◽  
Actuation Force ◽  
Force Sensitivity ◽  
Differential Operation ◽  
Static Equation ◽  
Pose Optimization

Abstract This paper proposes a novel performance index, which is called static actuation force sensitivity (SAFS), to investigate the response of the actuation forces when the amplitude of the suffered load of the end-effector has a change. Smaller SAFS can protect the actuations, and the load is mainly suffered by the structural constraints. This work starts with the construction of the unified forward Jacobian matrix of both serial and parallel mechanisms by screw theory. Then, with the forward Jacobian matrix, the inverse static equation is established. SAFS is thus introduced by the “partial differential” operation on the inverse static equation. SAFS is only related to the position of the whole mechanism and the direction of the suffered load, but not related to the detailed value of the amplitude of the load and the detailed value of the actuation forces; thus, SAFS can reveal the essence of static force capacities of the mechanisms. The example mechanism (namely, the 3revolute-prismatic-spherical (RPS) parallel mechanism) is used to illustrate the distribution of SAFS both over the workspace and at a certain pose. The analysis method of SAFS and the proposed index are expected to be applied to the pose optimization in the motion planning of the mechanisms to protect the actuations.

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