Forward Position Analysis of Nearly General Stewart Platforms

1994 ◽  
Vol 116 (1) ◽  
pp. 54-60 ◽  
Author(s):  
Chang-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

This paper presents the closed-form solution of the forward position analysis of the nearly general Stewart platform, which consists of a base and a moving planar platform 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, the coordinate 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 4th degree equations in three unknowns are derived. Further derivations produce 21 dependent constraint equations. By simultaneous elimination of two unknowns a 20th order polynomial equation in one unknown is obtained. Due to dual solutions of other unknowns, this indicates a maximum of 40 possible solutions. The roots of this polynomial are then solved numerically and the realistic solutions are constructed using computer graphics.

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.

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.

EVALUATION OF A NATURAL RUBBER SLEEVE STOPPER AS A CONTROLLED-RELEASE SUBSTRATE FOR USE IN STUDIES OF SEX-PHEROMONE-MEDIATED MATING DISRUPTION OF THE SPOTTED TENTIFORM LEAFMINER

1999 ◽  
Vol 131 (1) ◽  
pp. 137-149 ◽  
Author(s):  
R.M. Trimble ◽  
C.A. Tyndall ◽  
B.D. McGarvey
Keyword(s):  
Sex Pheromone ◽  
Natural Rubber ◽  
Mating Disruption ◽  
Polynomial Equation ◽  
Order Polynomial ◽  
Rubber Sleeve ◽  
Rate Of Evaporation ◽  
The Relationship ◽  
Linear Decline ◽  
Order Polynomial Equation

AbstractNatural rubber sleeve stoppers were impregnated with 10 mg of (E)-10-dodecen-1-yl-acetate, the major component of spotted tentiform leafminer, Phyllonorycter blancardella (F.), pheromone. In the laboratory, there was a linear decline in the amount of pheromone remaining on stoppers during 8 weeks of exposure to 10, 15, 20, and 25 °C. At 30 and 35 °C, the relationship was curvilinear and could be described using a second-order polynomial equation. The estimated rate of evaporation ranged from 0.03 mg/day at 10 °C to 0.08 mg/day at 25 °C; the rate of evaporation at 30 and 35 °C varied with the time since first exposure. There was a linear decline in the amount of pheromone remaining on stoppers during 8 weeks of exposure to fluctuating temperature regimes with average temperatures of 10 °C (i.e., 5–15 °C) and 20 °C (i.e., 15–25 °C). At a fluctuating regime with an average temperature of 30 °C (i.e., 25–35 °C), the relationship was curvilinear and could be described using a second-order polynomial equation. The estimated rate of evaporation was 0.02 and 0.09 mg/day at 5–15 and 15–25 °C, respectively; the estimated daily rate of evaporation at 25–35 °C varied with the time since first exposure. In an orchard, the estimated average rate of evaporation of pheromone from stoppers ranged from 0.05 to 0.31 mg/day and did not vary significantly with temperature. During the first 4 weeks of exposure in an orchard, the observed rate of evaporation was up to 4.4-fold greater than the rate predicted using the relationship between evaporation rate and constant temperatures observed in the laboratory. The potential for using natural rubber sleeve stoppers as controlled-release substrates in studies of sex-pheromone-mediated mating disruption of P. blancardella is discussed.

scholarly journals Digestibility, fermentation and microbiological characteristics of Calotropis procera silage with different quantities of grape pomace

2016 ◽  
Vol 40 (6) ◽  
pp. 698-705 ◽  
Author(s):  
Clérison dos Santos Belém ◽  
Anderson Miranda de Souza ◽  
Patrícia Rodrigues de Lima ◽  
Francisco Allan Leandro de Carvalho ◽  
Mário Adriano Ávila Queiroz ◽  
...  
Keyword(s):  
Lactic Acid ◽  
Animal Feed ◽  
Polynomial Equation ◽  
Grape Pomace ◽  
Calotropis Procera ◽  
Soluble Carbohydrate ◽  
In Vitro Digestibility ◽  
Fresh Matter ◽  
Order Polynomial ◽  
Order Polynomial Equation

ABSTRACT Preserving forage plants adapted to a semi-arid climate as silage may minimize the animal feed deficit during drought. The objective of this study was to evaluate the effects of different quantities of grape pomace added to Calotropis procera silage on its fermentation, in vitro digestibility, total digestible nutrients and microbiology. A completely randomized experimental design was used with four treatments (0, 10, 20 and 40% fresh matter) and four replicates. The silos were opened after 90 days of ensilage, and the soluble carbohydrate, ethanol, organic acid and ammoniacal nitrogen concentrations; pH; fermentation loss; dry matter (DM) recovery; DM density, and microbial populations were determined. The pH (3.96-3.87) was adequate for ensiling in all silage samples. The soluble carbohydrate concentration decreased (p<0.05), and the ethanol concentration increased with increasing quantities of grape pomace. The lactic acid concentration decreased (p<0.05) from 5.3 to 1.94% DM, and the acetic, propionic and butyric acid concentrations increased with increasing quantities of grape pomace. The lactic acid bacteria decreased linearly (p <0.05), varying from 6.43 to 5.82 log CFU/g silage. The mold and yeast population variations fit best using a third-order polynomial equation (p <0.05). Enterobacteria and Clostridium spp were not observed. Adding grape pomace to the silage increased the effluent and gas loss; the latter varied from 5.35 to 14.4%. The total digestible nutrient (TDN) variation fit best using a second-order polynomial equation, and the maximum value was estimated at 82.95% DM with 3.5% grape pomace using the regression equation. The percent digestibility decreased linearly (p<0.05) with increasing quantities of grape pomace. We show that Calotropis procera has potential as silage even without adding grape pomace.

Effects of Crystallization Degree on Rate during Isothermal Crystallization from PET Melt

2012 ◽  
Vol 174-177 ◽  
pp. 1520-1523
Author(s):  
Hong Ya Zhang ◽  
Feng Wang ◽  
Hua Cheng
Keyword(s):  
Heat Flow ◽  
Flow Rate ◽  
Crystallization Process ◽  
Crystallization Rate ◽  
Polynomial Equation ◽  
Heat Flow Rate ◽  
Order Polynomial ◽  
Coefficient Vector ◽  
Crystallization Degree ◽  
Order Polynomial Equation

In this study, one new function is defined as change of relative crystallization degree in unit time and named relative crystallization rate (1/min.). The curve of heat flow rate to time is transmitted to that of to . The produced curve was fitted using one high-order polynomial equation with a variable of and the coefficient vector （Ai ,in this paper，the values of i were from 0 to 9）was produced. It was found that, even during the accelerated stage of crystallization from PET melt，both aspects to promote and delay the relative crystallization rate existed， furthermore， both aspects of promotion and delay declined with the crystallization process and appeared “internal exhaustion”.

Soil-line vegetation indices for corn nitrogen content prediction

2012 ◽  
Vol 26 (2) ◽  
pp. 103-108 ◽  
Author(s):  
N. Bagheri ◽  
H. Ahmadi ◽  
S. Alavipanah ◽  
M. Omid
Keyword(s):  
Nitrogen Content ◽  
Vegetation Indices ◽  
Polynomial Equation ◽  
Measured Data ◽  
Second Order ◽  
Soil Line ◽  
Order Polynomial ◽  
Canopy Nitrogen ◽  
Order Polynomial Equation ◽  
Corn Canopy

Soil-line vegetation indices for corn nitrogen content prediction The soil-line vegetation indices for prediction of corn canopy nitrogen content were investigated. Results indicated that the vegetation indices applied were correlated with corn canopy nitrogen content and the wavelengths between 630-860 nm are suitable for nitrogen diagnosis. The second-order polynomial equation was the best model for nitrogen content prediction among different regression types. Analyses based on both predicted and measured data were carried out to compare the performance of existing vegetation indices.

Forward displacement analysis of a new 1CCC–5SPS parallel mechanism using Gröbner theory

2009 ◽  
Vol 223 (5) ◽  
pp. 1233-1241 ◽  
Author(s):  
D Gan ◽  
Q Liao ◽  
J S Dai ◽  
S Wei ◽  
L D Seneviratne
Keyword(s):  
Parallel Mechanism ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Geometrical Constraint ◽  
Degree Polynomial ◽  
Forward Displacement ◽  
Constraint Equations ◽  
Moving Platform ◽  
Forward Kinematic ◽  
Forward Displacement Analysis

A new parallel mechanism 1CCC–5SPS which has distance and angle constraints is introduced in this article. Degree of freedom and forward kinematic analysis of this new parallel mechanism are presented, in which four equivalent polynomial equations are obtained from the original six geometrical constraint equations. The Gröbner basis theory is used with the four equations and the problem of forward displacement is reduced to a 40th degree polynomial equation in a single unknown from a constructed 10 × 10 Sylvester's matrix which is small in size, from which 40 different locations of the moving platform can be derived. A numerical example confirms the efficiency of the procedure.

The Position Analysis of Spherical Seven-Bar Mechanism

2011 ◽  
Vol 101-102 ◽  
pp. 193-196
Author(s):  
Zhao Feng Zhang ◽  
Zhi Huan Zhang
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Polynomial Equation ◽  
Constraint Equations ◽  
Position Analysis ◽  
Numerical Example ◽  
Location Parameters ◽  
Sylvester Resultant ◽  
Univariate Polynomial

In this paper, we turn plane seven-bar mechanism into spherical seven-bar mechanism, using quaternion to construct mathematical model for spherical seven-bar mechanism. Three constraint equations are obtained according to the angles constraint. Using Sylvester resultant elimination by two steps, a 32 degree univariate polynomial equation can be obtained. A numerical example confirms that analytical solutions of spherical seven-bar mechanism are 32 and with the help of Mathematic software to solve the location parameters.

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.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

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