Determination of Constant Orientation Workspace of a Stewart Platform by Geometrical Method

2015 ◽  
Vol 813-814 ◽  
pp. 997-1001 ◽  
Author(s):  
S. Gokul Narasimhan ◽  
R. Shrivatsan ◽  
K. Venkatasubramanian ◽  
Anjan Kumar Dash
Keyword(s):  
Algebraic Method ◽  
Parallel Manipulators ◽  
Stewart Platform ◽  
Closed Form Expression ◽  
Robot Design ◽  
Form Expression ◽  
Novel Method ◽  
Workspace Optimization ◽  
Orientation Workspace

Determination of workspace is one of the main considerations in the design of any robot since the workspace geometry is considered a fundamental issue for robot design. This also plays a crucial role in trajectory planning. Among parallel manipulators, 6-DOF Stewart platforms is the most researched and widely used robot. However, till date there is no closed form expression of workspace volume for Stewart platform. In this paper, a novel method is proposed to find out the workspace volume of Stewart platform. In this paper, individual workspace of each leg of the manipulator (P-U-S) is determined and then translated by a common distance towards their geometrical center thus generating constant orientation workspace. To determine the workspace volume, geometric intersection of the six spheres is computed. This results in workspace of definite shape and size, whose volume is calculated using simple formulae. It is observed that the geometric way of determination of workspace area is computationally less tedious than the algebraic method. This also helps a lot for workspace optimization of such manipulators.

A geometric approach for the workspace analysis of two symmetric planar parallel manipulators

Robotica ◽  
2014 ◽  
Vol 34 (4) ◽  
pp. 738-763 ◽  
Author(s):  
Banke Bihari ◽  
Dhiraj Kumar ◽  
Chandan Jha ◽  
Vijay S. Rathore ◽  
Anjan Kumar Dash
Keyword(s):  
Optimum Design ◽  
Critical Parameter ◽  
Geometric Approach ◽  
Parallel Manipulators ◽  
Closed Form Expression ◽  
Form Expression ◽  
Workspace Analysis ◽  
Geometrical Relationship ◽  
Planar Parallel Manipulators

SUMMARYThe workspace is often a critical parameter for optimum design of parallel manipulators. Workspace shape and area are two important considerations under this. In this paper, 5-R and 3-RRR planar parallel manipulators having symmetric link lengths are considered for workspace analysis. Here, symmetric means that the lengths of the first and second links of the legs are the same in all branches. Workspace analysis for such manipulators is normally done in a nondimensional way. The determination of the workspace area is one of the important parameters in the optimum design of a manipulator, and the determination of the area in terms of nondimensional parameters is extremely difficult in the case of 3-DOF and higher-DOF manipulators. In this paper, a geometric method is presented to determine different workspace shapes and areas. Based on this, all possible shapes of workspace are presented for both 5-R and 3-RRR planar parallel manipulators. For each case, a geometrical relationship between the link lengths is determined. The geometric approach gives a closed-form expression for the area calculation, which is not possible when adopting a nondimensional approach. In addition, this approach provides relationships between workspace shape and area and link lengths.

Evaluation and Representation of the Theoretical Orientation Workspace of the Gough–Stewart Platform

2009 ◽  
Vol 1 (2) ◽  
Author(s):  
Qimi Jiang ◽  
Clément M. Gosselin
Keyword(s):  
Theoretical Orientation ◽  
Robotic Manipulators ◽  
Stewart Platform ◽  
Closed Region ◽  
Leg Length ◽  
Fixed Frame ◽  
Cartesian Space ◽  
Orientation Workspace

The evaluation and representation of the orientation workspace of robotic manipulators is a challenging task. This work focuses on the determination of the theoretical orientation workspace of the Gough–Stewart platform with given leg length ranges [ρimin,ρimax]. By use of the roll-pitch-yaw angles (ϕ,θ,ψ), the theoretical orientation workspace at a prescribed position P0 can be defined by up to 12 workspace surfaces. The defined orientation workspace is a closed region in the 3D orientation Cartesian space Oϕθψ. As all rotations R(x,ϕ), R(y,θ), and R(z,ψ) take place with respect to the fixed frame, any point of the defined orientation workspace provides a clear measure for the platform to, respectively, rotate in order around the (x,y,z) axes of the fixed frame. An algorithm is presented to compute the size (volume) of the theoretical orientation workspace and intersectional curves of the workspace surfaces. The defined theoretical orientation workspace can be applied to determine a singularity-free orientation workspace.

Conditions for Translation and Rotation of Bodies on Frictional Surfaces

1991 ◽  
Vol 205 (5) ◽  
pp. 311-316
Author(s):  
D Nowell ◽  
D A Hills ◽  
R L Munisamy
Keyword(s):  
Closed Form ◽  
Numerical Technique ◽  
Closed Form Expression ◽  
Frictional Surface ◽  
Form Expression ◽  
Straight Line ◽  
Frictional Surfaces

This paper is concerned with the quasi-static motion of a body resting on a frictional surface when subjected to an arbitrary imposed displacement by a small frictionless finger. A simplified object with three distinct feet is used as an illustration but the method adopted can be generalized to bodies with more feet or distributed contact. A closed-form expression is found which enables the determination of the conditions necessary for a body to rotate about a point of support and a more general numerical technique for determining the instantaneous centre is presented. The indeterminacy that occurs when the points of support lie in a straight line is also discussed.

Evaluation and Representation of the Orientation Workspace of the Gough-Stewart Platform

2008 ◽  
Author(s):  
Qimi Jiang ◽  
Cle´ment M. Gosselin
Keyword(s):  
Numerical Algorithm ◽  
Robotic Manipulators ◽  
Stewart Platform ◽  
Leg Length ◽  
Fixed Frame ◽  
Orientation Space ◽  
Orientation Workspace

The evaluation and representation of the orientation workspace of robotic manipulators is a challenging task. This work focuses on the determination of the orientation workspace of the Gough-Stewart platform with given leg length ranges [ρimin, ρimax]. By use of the Roll–Pitch–Yaw angles (φ, θ, ψ), the orientation workspace at a prescribed position can be defined by 12 workspace surfaces. The obtained orientation workspace is a region in the 3D Cartesian orientation space O φ θ ψ. As all rotations R(x, φ), R(y, θ) and R(z, ψ) take place with respect to the fixed frame, any point of the orientation workspace provides a clear measure for the platform to respectively rotate in order around the (x, y, z) axes of the fixed frame. Also, as the shape of the 3D orientation workspace is very complex, a numerical algorithm is presented to compute its volume.

Rotation–Vibration Coupling in Diatomic Molecules and the Factorization Method. II. Closed Form Formulas for the Morse–Pekeris Intensities

1974 ◽  
Vol 52 (2) ◽  
pp. 110-119 ◽  
Author(s):  
M. Badawi ◽  
N. Bessis ◽  
G. Bessis ◽  
G. Hadinger
Keyword(s):  
Dipole Moment ◽  
Closed Form ◽  
Explicit Expression ◽  
Factorization Method ◽  
Closed Form Expression ◽  
Matrix Elements ◽  
Moment Matrix ◽  
Form Expression ◽  
Vibration Coupling

It is shown that, by applying an "accelerated" ladder operatorial formalism or an equivalent matrix procedure, one can obtain, easily, for the case of a Morse–Pekeris potential, a closed form expression of the rotation–vibration nuclear dipole moment matrix elements. This explicit expression, which is valid for any degree k of the dipole moment Taylor's expansion, allows the determination of the rotation–vibration intensities for any ΩνJ → Ω′ν′J′ transition.

scholarly journals Approximation of Capacity for Downlink Multi-User System with Combination of Precoding and NOMA Methods

2021 ◽  
Vol 11 (22) ◽  
pp. 10578
Author(s):  
Vu Van Son ◽  
Nguyen Le Cuong ◽  
Nguyen Thu Phuong ◽  
Tran Manh Hoang ◽  
Pham Thanh Hiep
Keyword(s):  
Interference Cancellation ◽  
Multiple Access ◽  
Wireless Systems ◽  
Ergodic Capacity ◽  
Closed Form Expression ◽  
Important Research ◽  
Analysis Method ◽  
Form Expression ◽  
Access Method ◽  
Novel Method

Enhancing performance of downlink MU systems is an attractive and important research for future wireless systems. The non-orthogonal multiple access (NOMA) method was proposed to improve the performance of MU systems. In order to further improve the outage probability (OP) and ergodic capacity (EC) of downlink NOMA MU systems, we propose the combination of precoding and NOMA methods, and then the OP and EC of MU systems with our novel method are derived in scenarios of perfect and imperfect successive interference cancellation (SIC) scheme. Moreover, the closed-form expression of OP and EC for both scenarios is theoretically derived and compared with Monte Carlo simulations. The results show that, the analysis method is accurate, and the proposed combining precoding and NOMA can further enhance the performance of MU systems in comparing with the original orthogonal multiple access method.

SIMPA: A Graphic Simulator for the CAD of Parallel Manipulators

1992 ◽  
Author(s):  
Clément M. Gosselin ◽  
Simon Laverdière ◽  
Jean Côte
Keyword(s):  
Graphical Representation ◽  
Mechanical Design ◽  
Parallel Manipulators ◽  
Mechanical Analysis ◽  
Complex Issue ◽  
Planar Manipulators ◽  
Stiffness Properties ◽  
Contour Plots ◽  
Workspace Optimization

Abstract This paper presents a graphic simulation environment which can be used as an aid for the design of parallel manipulators. Both planar and spatial parallel manipulators can be treated with the simulator. Fast algorithms for the determination of the workspace are implemented. Therefore, sections of the workspace — with their boundaries — can be plotted interactively and their volume or area estimated very accurately. Moreover, contour plots of design criteria such as dexterity, stiffness or conditioning can be superimposed on these plots in order to provide the user with a graphical representation of the behavior of these properties over the workspace. Singularity loci can also be plotted for planar manipulators. The simulator is implemented on a SUN SPARC Station and uses the SUNPHIGS graphics library. The mechanical design of parallel manipulators is a complex issue which involves a detailed mechanical analysis. Indeed, parallel manipulators generally exhibit good stiffness properties but usually have a rather limited workspace. Furthermore, they are known to suffer from singularities for which the end-effector becomes uncontrollable. Problems related to workspace optimization and stiffness are therefore crucial in the context of design of parallel manipulators and the simulation package presented here is a very useful tool for the designer.

Sign in / Sign up

Export Citation Format

Share Document