A Real Parameter Continuation Method for Complete Solution of Forward Position Analysis of the General Stewart

2002 ◽  
Vol 124 (2) ◽  
pp. 236-244 ◽  
Author(s):  
Zongliang Mu ◽  
Kazem Kazerounian
Keyword(s):  
Numerical Method ◽  
Complete Solution ◽  
Continuation Method ◽  
Stewart Platform ◽  
Target System ◽  
Analysis Problem ◽  
Position Analysis ◽  
Six Degree Of Freedom ◽  
Target Platform ◽  
Base Platform

Stewart Platform is a six degree of freedom, parallel manipulator, which consists of a base platform, a coupler platform and six limbs connected at six distinct points on the base platform and the coupler platform. The forward position analysis problem of Stewart Platform amounts to finding all its possible configurations based on the knowledge of the lengths of its limbs. In this paper, we present a numerical method for solving the forward position analysis problem for the most general Stewart Platform. This is a numerical method based on the polynomial continuation as established in recent works in the literature. However, one main difference is that the start system and the homotopy used here are based on physical design rather than pure mathematical equations. First, the target Stewart Platform is geometrically simplified into a platform, which, as the start platform, can be solved analytically. Then, a homotopy is constructed between the kinematics equations of the start platform and those of the target platform. By changing the parameters of the start platform incrementally into the parameters of the target system while tracking solutions of the start platform, a complete set of 40 solutions to the target platform can be found. Through this process, all of the extraneous paths have been eliminated before the solution tracking procedure starts and only isolated solutions of the start platform are tracked. The process for solutions to switch between real and complex is examined.

An Algebraic Method for Direct Position Analysis of the General Stewart Platform Manipulator Robot

2009 ◽  
Vol 626-627 ◽  
pp. 405-410
Author(s):  
Xi Guang Huang ◽  
Guang Pin He ◽  
Q.Z. Liao
Keyword(s):  
Parallel Manipulator ◽  
Algebraic Method ◽  
Stewart Platform ◽  
Mobile Platform ◽  
Analysis Problem ◽  
Position Analysis ◽  
General Geometry ◽  
Six Degree Of Freedom ◽  
Manipulator Robot ◽  
Base Platform

Stewart platform manipulator robot is a six degree of freedom, parallel manipulator, which consists of a base platform, a mobile platform and six limbs connected at six distinct points on the base platform and the mobile platform respectively. The direct position analysis problem of Stewart platform relates to the determination of the mobile platform pose for a given set of the lengths of the limbs. In this paper, we present a concise algebraic method for solving the direct position analysis problem for the fully parallel manipulator with general geometry, often referred to as General Stewart platform manipulator. Based on the presented algebraic method, we derive a 40th degree univariate polynomial from a determinant of 20×20 Sylvester’s matrix, which is relatively small in size. We also obtain a complete set of 40 solutions to the most general Stewart platform. The proposed method is comparatively concise and reduces the computational burden. Finally the method is demonstrated by a numerical example.

Six Degree of Freedom Active Isolation in Flexible Supporting Structures: Numerical Simulation and Experiment

2003 ◽  
Author(s):  
Yuan Cheng ◽  
Qian Zhou ◽  
Ge-Xue Ren ◽  
Hui Zhang
Keyword(s):  
Stewart Platform ◽  
Degree Of Freedom ◽  
Active Isolation ◽  
Dynamics And Control ◽  
Six Degree Of Freedom ◽  
Multi Body ◽  
And Control ◽  
Flexible Cables ◽  
Base Platform ◽  
Supporting Structures

This paper studies the six degree-of-freedom active isolation of flexible supporting structures using Gough-Stewart platform. The problem arises from a large radio telescope in which the astronomical equipment is mounted on a platform to be stabilized, while the base platform of the mechanism itself is carried by a cable car moving along flexible cables. In this paper, the stabilization problem is equivalent to a dynamics and control problem of multi-body system. A control law of the prediction of the base platform and PD feedback is proposed for the six actuators of the Gough-Stewart platform. Based on numerical results, a model experimental setup has been built up. The control effects are measured with LTD 500 Laser Tracker.

scholarly journals Coupler-Point-Curve Synthesis Using Homotopy Methods

1990 ◽  
Vol 112 (3) ◽  
pp. 384-389 ◽  
Author(s):  
Lung-Wen Tsai ◽  
Jeong-Jang Lu
Keyword(s):  
Numerical Method ◽  
Continuation Method ◽  
Homotopy Method ◽  
Synthesis Problem ◽  
Singular Solutions ◽  
Homotopy Methods ◽  
Point Curve ◽  
Non Linear

A numerical method called “Homotopy Method” (or Continuation Method) is applied to the problem of four-bar coupler-curve synthesis. We have shown that: for five precision points, the “General Homotopy Method” can be applied to find the link lengths of number of four-bar linkages, and for nine precision points, a heuristic “Cheater’s Homotopy” can be applied to find some four-bar linkages. The nine-coupler-points synthesis problem is highly non-linear and highly singular. We have found that Newton-Raphson’s method and Powell’s method tend to converge to the singular solutions or do not converge at all, while the Cheater’s Homotopy always finds some non-singular solutions although sometimes the solutions may be complex.

Forward Position Analysis of Nearly General Stewart Platforms

1992 ◽  
Author(s):  
Change-de Zhang ◽  
Shin-Min Song
Keyword(s):  
Closed Form Solution ◽  
Polynomial Equation ◽  
Stewart Platform ◽  
Form Solution ◽  
Constraint Equations ◽  
Position Analysis ◽  
Order Polynomial ◽  
Moving Platform ◽  
Order Polynomial Equation ◽  
Simultaneous Elimination

Abstract This paper presents the closed-form solution of forward position analysis of the nearly general stewart platform, which consists of a base and a moving planar platforms connected by six extensible limbs through spherical joints in the two planar platforms. It becomes a general stewart platform if the centers are not constrained to those two planes. In this study, transformation matrix is used to represent the position of the moving platform. Based on the six dependency equations of the rotation matrix and the six constraint equations related to the six link lengths, a set of six 4-th degree equations in three unknowns are derived. Further derivations produce twenty-one dependent constraint equations. By simultaneous elimination of two unknowns a 20-th order polynomial equation in one unknown is obtained. Due to dual solutions of other unknowns, this indicates a maximum of forty possible solutions. The roots of this polynomial are solved numerically and the realistic solutions are constructed using computer graphics.

In-Plane Wave Propagation Through Elastic Solids With a Periodic Array of Rectangular Defects

2001 ◽  
Vol 69 (2) ◽  
pp. 179-188 ◽  
Author(s):  
E. Scarpetta ◽  
M. A. Sumbatyan
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Numerical Method ◽  
Analytical Approach ◽  
Complete Solution ◽  
Periodic Array ◽  
Elastic Solids ◽  
Longitudinal Plane ◽  
Direct Numerical Method ◽  
Plane Wave Propagation

In the context of wave propagation in damaged (elastic) solids, an analytical approach is developed to study normal penetration of a longitudinal plane wave into a periodic array of rectangular defects. Reducing the problem to some integral equations holding over the base and height of the openings, a direct numerical method is applied to give a complete solution for various exact or approximated forms. Several figures show the peculiarities of the structure and lead to physical conclusions.

A New Numerical Method for Simulation

SIMULATION ◽  
1965 ◽  
Vol 4 (5) ◽  
pp. 324-330 ◽  
Author(s):  
Maury E. Fowler
Keyword(s):  
Computer Program ◽  
Numerical Method ◽  
Degree Of Freedom ◽  
Root Locus ◽  
Guidance And Control ◽  
Bending Modes ◽  
Six Degree Of Freedom ◽  
And Control

This paper illustrates a method of digitally simulating the motion and control of vehicles. This method of simulation has been used for the simulation of a large variety of systems, including a six-degree-of- freedom simulation of Gemini reentry with guidance and control, a six-degree-of-freedom aircraft simula tion, missile control simulations including bending modes, trajectory problems, and many others. The major advantage of the method, in addition to the speed of computation, is that a detailed an alysis of the system is carried out during the develop ment of the simulation. Although, for small problems, simulations using this method can be derived by hand, large simula tions generally require a computer program to com pute root locus points and z-transforms.

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