Dimensional Mobility Criteria of Planar 6T-9R Paradoxical Chains

2012 ◽  
Author(s):  
Chung-Ching Lee ◽  
Jeng-Hsiung Lee ◽  
Po-Chih Lee
Keyword(s):  
Structural Type ◽  
Degree Of Freedom ◽  
Sixth Order ◽  
Constrained Motion ◽  
Numerical Examples ◽  
Euclidean Metric ◽  
Order Polynomial ◽  
Depth Study

Focusing on the structural type of 6 ternary links and 9 revolute pairs (denoted 6T-9R), which stems from the commercialized Expanda-Triangles construction toy, we conduct in-depth study on the mobility constraints of the generalized planar 6T-9R paradoxical chains. According to Grübler criterion, the degree of freedom (DoF) of the chains of this type is minus 3 and they should have no constrained motion. However, due to the special dimensional constraints (Euclidean metric), some of such paradoxical mechanisms can still have the full-cycle mobility. To identify the dimensional constraints of mobility algebraically, we adopt a simple direct algebraic elimination approach to attain an eliminant of sixth-order polynomial in one variable only. The identity of polynomial whose coefficients are merely function of link lengths and structural angles produces six necessary dimensional constraints for movable 6T-9R paradoxical chain. One new and two existing paradoxical chains are revealed and their numerical examples are illustrated to show the correctness and validity of the proposed dimensional constraints. A novel generalized form of planar 6-bar paradoxical chain is disclosed too.

Forward Position Analysis of Two 3-R Robots Manipulating a Planar Linkage Payload

1995 ◽  
Vol 117 (2A) ◽  
pp. 292-297 ◽  
Author(s):  
G. R. Pennock ◽  
K. G. Mattson
Keyword(s):  
Closed Loop ◽  
Degree Of Freedom ◽  
Sixth Order ◽  
Position Analysis ◽  
Revolute Joints ◽  
Order Polynomial ◽  
Cooperating Robots ◽  
Position Problem ◽  
Three Degree Of Freedom ◽  
Insight Into

This paper presents the forward position analysis of two planar three degree-of-freedom robots, with all revolute joints, manipulating a single degree-of-freedom closed-loop linkage payload. Kinematic constraint relations are developed which provide geometric insight into the cooperating robot-payload system and are important in the control of the two robots. For illustrative purposes, the payload that is considered here is a planar four-bar linkage. The paper shows that the orientation of a specified link in the payload can be described by a sixth-order polynomial. This polynomial is an important contribution, not only to the kinematics of the cooperating robots, but to the multiple-input closed-loop nine-bar linkage formed by the two robots and the payload. The polynomial contains important information regarding the assembly configurations and the stationary configurations of the system. The paper shows that zero, two, four, or six assembly configurations are possible and that each configuration corresponds to a different circuit of the system. Graphical methods are utilized to provide geometric insight into the assembly and stationary configurations and to check the results obtained from the sixth-order polynomial. A numerical example is included which demonstrates the importance of the polynomial in solving the forward position problem, and in determining the number of assembly configurations.

Direct Differential Kinematics of Hybrid-Chain Manipulators Including Singularity and Stability Analyses

1994 ◽  
Vol 116 (2) ◽  
pp. 614-621 ◽  
Author(s):  
Yong-Xian Xu ◽  
D. Kohli ◽  
Tzu-Chen Weng
Keyword(s):  
Stability Index ◽  
Inverse Kinematic ◽  
Degree Of Freedom ◽  
Static Force ◽  
Force Analysis ◽  
Kinematic Singularity ◽  
Numerical Examples ◽  
Transformation Matrices ◽  
Unstable Configuration ◽  
Direct Kinematics

A general formulation for the differential kinematics of hybrid-chain manipulators is developed based on transformation matrices. This formulation leads to velocity and acceleration analyses, as well as to the formation of Jacobians for singularity and unstable configuration analyses. A manipulator consisting of n nonsymmetrical subchains with an arbitrary arrangement of actuators in the subchain is called a hybrid-chain manipulator in this paper. The Jacobian of the manipulator (called here the system Jacobian) is a product of two matrices, namely the Jacobian of a leg and a matrix M containing the inverse of a matrix Dk, called the Jacobian of direct kinematics. The system Jacobian is singular when a leg Jacobian is singular; the resulting singularity is called the inverse kinematic singularity and it occurs at the boundary of inverse kinematic solutions. When the Dk matrix is singular, the M matrix and the system Jacobian do not exist. The singularity due to the singularity of the Dk matrix is the direct kinematic singularity and it provides positions where the manipulator as a whole loses at least one degree of freedom. Here the inputs to the manipulator become dependent on each other and are locked. While at these positions, the platform gains at least one degree of freedom, and becomes statically unstable. The system Jacobian may be used in the static force analysis. A stability index, defined in terms of the condition number of the Dk matrix, is proposed for evaluating the proximity of the configuration to the unstable configuration. Several illustrative numerical examples are presented.

Dynamics of a Two-DOF Parallel Pointing Mechanism

2005 ◽  
Author(s):  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Numerical Simulation ◽  
Parallel Mechanism ◽  
Inverse Dynamics ◽  
Kinematic Model ◽  
Degree Of Freedom ◽  
Numerical Examples ◽  
Proposed Model ◽  
Dynamic Analyses ◽  
Moving Platform ◽  
Two Degree Of Freedom

This paper presents kinematic and dynamic analyses of a two-degree-of-freedom pointing parallel mechanism. The mechanism consists of a moving platform, connected to a fixed platform by two legs of type PUS (prismatic-universal-spherical). At first a simplified kinematic model of the pointing mechanism is introduced. Based on this proposed model, the dynamics equations of the system using the Natural Orthogonal Complement method are developed. Numerical examples of the inverse dynamics results are presented by numerical simulation.

A Technique for Determining Damping Values and Damper Locations in Multi-Degree-of-Freedom Systems

1975 ◽  
Vol 97 (4) ◽  
pp. 1245-1250 ◽  
Author(s):  
J. N. Kanianthra ◽  
F. H. Speckhart
Keyword(s):  
Degree Of Freedom ◽  
Numerical Examples ◽  
Linear Damping

In complicated multi-degree-of-freedom systems, it is often necessary to choose linear damping values and determine their locations that will give a desired performance. The method presented in this paper describes procedures by which the damping values and damper locations for prescribed damping ratios in all principal modes can be found. Two numerical examples are included to illustrate the method.

Analytic Geometric Design of Spatial R-R Robot Manipulators

2000 ◽  
Author(s):  
Constantinos Mavroidis ◽  
Munshi Alam ◽  
Eric Lee
Keyword(s):  
Degrees Of Freedom ◽  
Robot Manipulators ◽  
Open Loop ◽  
Geometric Design ◽  
Sixth Order ◽  
Design Parameters ◽  
End Effector ◽  
Revolute Joints ◽  
Order Polynomial ◽  
Spatial Locations

Abstract This paper studies the geometric design of spatial two degrees of freedom, open loop robot manipulators with revolute joints that perform tasks, which require the positioning of the end-effector in three spatial locations. This research is important in situations where a robotic manipulator or mechanism with a small number of joint degrees of freedom is designed to perform higher degree of freedom end-effector tasks. The loop-closure geometric equations provide eighteen design equations in eighteen unknowns. Polynomial Elimination techniques are used to solve these equations and obtain the manipulator Denavit and Hartenberg parameters. A sixth order polynomial is obtained in one of the design parameters. Only two of the six roots of the polynomial are real and they correspond to two different robot manipulators that can reach the desired end-effector poses.

scholarly journals Stationary values of the transmission ratio of the planar fourbar

2008 ◽  
Vol 30 (4) ◽  
Author(s):  
J. Wittenburg
Keyword(s):  
Polynomial Equation ◽  
Sixth Order ◽  
New Method ◽  
Transmission Ratio ◽  
Output Link ◽  
Order Polynomial ◽  
Angular Velocities ◽  
Order Polynomial Equation

The transmission ratio of the planar fourbar, i.e. the ratio of the angular velocities of input link and output link, is a function of the input angle. Freudenstein [1] showed how to calculate stationary values of the transmission ratio. In the present paper a new method is described. Like Freudenstein’s method it results in a sixth-order polynomial equation.

Passive Control Analysis and Design of Twin-Tower Structure with Chassis

2020 ◽  
Vol 20 (06) ◽  
pp. 2040010
Author(s):  
Qiaoyun Wu ◽  
Hai Feng ◽  
Shiye Xiao ◽  
Hongping Zhu ◽  
Xixuan Bai
Keyword(s):  
Minimum Energy ◽  
Degree Of Freedom ◽  
Vibration Energy ◽  
Control Analysis ◽  
Numerical Examples ◽  
Analysis And Design ◽  
Mdof Systems ◽  
Tower Structure ◽  
Passive Dampers ◽  
Analytical Expressions

In this paper, a symmetrical twin-tower structure with chassis connected with passive dampers is coupled as 2-DOF (degree of freedom) model. Using the stationary white noise as seismic excitation, the frequency–response function and the vibration energy expression of the symmetrical twin-tower structure are established based on the simplified 2-DOF model. Furthermore, based on the principle of minimum energy, the analytical expressions of the optimization parameters of two kinds of passive dampers are deduced, and the effectiveness of the dampers with optimized coefficients on structural control is verified by numerical examples of 2-DOF and MDOF (multi-degree-of-freedom) systems, respectively. Finally, the control effects of the two kinds of dampers under different control strategies on the responses of displacement of the top, base shear, structural vibration energy, and maximum inter-story drift of the symmetrical twin-tower structure are discussed through three-dimensional finite element numerical examples. It is verified that the analytical expressions of optimum parameters of the two kinds of dampers proposed based on the 2-DOF model are also beneficial to reduce the responses of the MDOF systems and actual engineering.

scholarly journals Some novel Newton-type methods for solving nonlinear equations

2019 ◽  
Vol 38 (3) ◽  
pp. 111-123
Author(s):  
Morteza Bisheh-Niasar ◽  
Abbas Saadatmandi
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Sixth Order ◽  
New Method ◽  
Order Convergence ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Newton Iterative Method

The aim of this paper is to present a new nonstandard Newton iterative method for solving nonlinear equations. The convergence of the proposed method is proved and it is shown that the new method has cubic convergence. Furthermore, two new multi-point methods with sixth-order convergence, based on the introduced method, are presented. Also, we describe the basins of attraction for these methods. Finally, some numerical examples are given to show the performance of our methods by comparing with some other methods available in the literature

scholarly journals Enriched degree of freedom locking in crack analysis using the extended finite element method

2019 ◽  
Vol 11 (6) ◽  
pp. 168781401985979
Author(s):  
Han-Soo Kim ◽  
Geon-Hyeong Kim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Extended Finite Element Method ◽  
Degree Of Freedom ◽  
Analysis Model ◽  
Extended Finite Element ◽  
Simple Method ◽  
Numerical Examples ◽  
Crack Analysis ◽  
Element Method

In this article, the enriched degree of freedom locking that can occur in a crack analysis with the extended finite element method is described. The discontinuous displacement field formulated by the enriched degree of freedom in the extended finite element method does not activate due to the enriched degree of freedom locking. Using the phantom node method, the occurrence of locking when two adjacent elements are simultaneously cracked in a loading step was verified. Two adjacent cracks can be determined to have developed simultaneously when an analysis model reveals a relatively uniform stress distribution on two adjacent elements. Numerical examples of a simply tensioned bar and a reinforced concrete beam are presented to demonstrate the erroneous analysis result due to the enriched degree of freedom locking. As a simple method to circumvent the enriched degree of freedom locking, the tensile strength of the neighboring elements was slightly increased in the numerical examples, and the effectiveness of the method was demonstrated. The proposed method is simple and easy for practicing engineers, and it can be easily applied to the three-dimensional crack propagation analysis.

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