Direct Position Kinematics of the 3-1-1-1 Stewart Platforms

1992 ◽  
Author(s):  
Muqtada Husain ◽  
Kenneth J. Waldron
Keyword(s):  
Closed Form ◽  
Special Class ◽  
General Feature ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Numerical Example ◽  
Geometric Argument ◽  
Stewart Platforms ◽  
Base Platform

Abstract In this work, a closed form solution for the direct position kinematics problem of a special class of Stewart Platform is presented. This class of mechanisms has a general feature that the top platform is connected to the six limbs at four locations. Three limbs connect at one location and the remaining limbs connect to the top platform singly at three separate locations. The base platform is connected at six different locations as is the case in the general platform. This particular class of mechanism is termed as 3-1-1-1 mechanism in this paper. It has been shown that there are a maximum of sixteen real assembly configurations for the direct position kinematics problem. This has been verified using a geometric argument also. The numerical example solved in this paper demonstrates that it is possible to obtain a set of solutions which are all real.

Direct Position Kinematics of the 3-1-1-1 Stewart Platforms

1994 ◽  
Vol 116 (4) ◽  
pp. 1102-1107 ◽  
Author(s):  
M. Husain ◽  
K. J. Waldron
Keyword(s):  
Special Class ◽  
General Feature ◽  
Stewart Platform ◽  
Numerical Example ◽  
Geometric Argument ◽  
Stewart Platforms ◽  
Base Platform

In this work, a closed from solution for the direct position kinematics problem of a special class of Stewart Platform is presented. This class of mechanisms has a general feature that the top platform is connected to the six limbs at four locations. Three limbs connect at one location and the remaining limbs connect to the top platform singly at three separate locations. The base platform is connected at six different locations as is the case in the general platform. This particular class of mechanism is termed as 3-1-1-1 mechanism in this paper. It has been shown that there are a maximum of sixteen real assembly configurations for the direct position kinematics problem. This has been verified using a geometric argument also. The numerical example solved in this paper demonstrates that it is possible to obtain a set of solutions which are all real.

Direct Position Analysis of the 4–6 Stewart Platforms

1994 ◽  
Vol 116 (1) ◽  
pp. 61-66 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Although Stewart platforms have been applied in the design of aircraft and vehicle simulators and parallel robots for many years, the closed-form solution of direct (forward) position analysis of Stewart platforms has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral, the 3–6 and the 4–4 Stewart platforms, of which the forward position solutions were derived as an eighth or a twelfth degree polynomials with one variable in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more general case of the Stewart platform, the 4–6 Stewart platforms, in which two pairs of the upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of the 3–6 and 4–4 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

scholarly journals A Closed-Form Solution for the Boundary Value Problem of Gas Pressurized Circular Membranes in Contact with Frictionless Rigid Plates

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1017
Author(s):  
Dong Mei ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Circular Membrane ◽  
Rigid Plate ◽  
Pressure Loading ◽  
Numerical Example

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify.

Structural Error Synthesis of a Four-Bar Function Generator Using the Reliability Concept

1984 ◽  
Vol 198 (2) ◽  
pp. 87-90 ◽  
Author(s):  
A K Khare ◽  
A C Rao
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimization Techniques ◽  
Function Generator ◽  
New Approach ◽  
Numerical Example ◽  
Structural Error ◽  
Synthesis Of Mechanisms

Structural error synthesis of mechanisms is usually carried out either by the precision point approach or by using optimization techniques. A new approach for such problems using the reliability concept is presented in this paper. Besides being simple, this approach leads to a closed form solution and the mechanism can be designed to perform with any desired reliability. Its application is illustrated by means of a numerical example and the results are compared with those available.

Closed-form solution of the direct kinematics of the 6–3 type Stewart Platform using one extra sensor

Meccanica ◽  
1996 ◽  
Vol 31 (6) ◽  
pp. 705-714 ◽  
Author(s):  
Vincenzo Parenti-Castelli ◽  
Raffaele Di Gregorio
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Direct Kinematics

Direct Position Analysis of the 4-6 Stewart Platforms

1992 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Unknown Variable ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Abstract Although Stewart platforms have been applied in the designs of aricraft and vehicle simulators and parallel robots in many years, their closed-form solution of direct (forward) position analysis has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral Stewart and the 4-4 Stewart platforms, in which two pairs of both upper and lower joint centers are coincident. The former results in in an eighth degree polynomial and the latter results in an eighth and a twelfth degree polynomials for different cases. The single unknown variable is in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more genearl case of the Stewart platform, the 4-6 Stewart platforms, in which two pairs of upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of 4-4 and and 3-6 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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