Solution of the Direct Position Kinematics Problem of the General Stewart Platform Using Advanced Polynomial Continuation

1992 ◽  
Author(s):  
S. V. Sreenivasan ◽  
Prabjot Nanua
Keyword(s):  
Closed Form ◽  
Numerical Algorithm ◽  
Polynomial System ◽  
Stewart Platform ◽  
Geometric Parameters ◽  
Polynomial Equations ◽  
Number Of Solutions ◽  
Closed Form Solutions ◽  
General Geometry ◽  
First Time

Abstract In this paper the direct position kinematics problem of the most general geometry of the Stewart platform has been solved numerically. It is found that this problem has exactly 40 distinct, finite solutions. No closed form solutions are available for this problem. This problem was addressed for the first time by Raghavan recently, and the results obtained here are in agreement with the ones obtained by Raghavan. The formulation and the variable set used in this work lead to a simpler polynomial system of lower overall degree. Further, the geometric parameters of the general Stewart platform have been used to generate a ‘near-general’ Stewart platform. Even though the near-general Stewart platform also has exactly 40 finite, distinct, configurations, it is represented by much simpler polynomial equations. The polynomial system obtained from this near-general Stewart platform is used as the start system in an advanced polynomial continuation scheme to solve the general Stewart problem. This process eliminates a large number of solutions at infinity that have no geometric significance. This leads to an efficient numerical algorithm that is about twenty times faster than traditional continuation schemes.

scholarly journals Delay chemical master equation: direct and closed-form solutions

2015 ◽  
Vol 471 (2179) ◽  
pp. 20150049 ◽  
Author(s):  
Andre Leier ◽  
Tatiana T. Marquez-Lago
Keyword(s):  
Master Equation ◽  
Closed Form ◽  
Stochastic Simulation Algorithm ◽  
Chemical Master Equation ◽  
Kolmogorov Equation ◽  
Closed Form Solutions ◽  
Mrna Maturation ◽  
First Time ◽  
Nonlinear Markov Process ◽  
Reaction Schemes

The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

Use of Resultants and the Bernshtein Formula in the Dimensional Synthesis of Linkage Components

1994 ◽  
Vol 116 (4) ◽  
pp. 1171-1172 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Fourth Degree ◽  
Finite Precision ◽  
Closed Form Solutions ◽  
System Of Polynomial Equations ◽  
Resultant Theory

The applications of resultants and the Bernshtein formula for the dimensional synthesis of linkage components for finite precision positions are discussed. The closed-form solutions, which are derived from systems of polynomials in multiple unknowns by applying resultant theory, are in forms of polynomial equations of a single unknown. For the case of two compatibility equations, the closed form solution is a sixth degree solution polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree solution polynomial. For each case, the Bernshtein formula is applied to calculate the number of solutions of the system of polynomial equations. The calculated numbers of solutions match the degrees of the solution polynomials for both cases.

KINEMATIC ANALYSIS AND PROTOTYPE OF A METAMORPHIC ANTHROPOMORPHIC HAND WITH A RECONFIGURABLE PALM

2011 ◽  
Vol 08 (03) ◽  
pp. 459-479 ◽  
Author(s):  
GUOWU WEI ◽  
JIAN S. DAI ◽  
SHUXIN WANG ◽  
HAIFENG LUO
Keyword(s):  
Closed Form ◽  
Matrix Equation ◽  
Kinematic Analysis ◽  
Structure Design ◽  
Characteristic Matrix ◽  
Robotic Hand ◽  
Closed Form Solutions ◽  
First Time

A novel metamorphic anthropomorphic hand is for the first time introduced in this paper. This robotic hand has a reconfigurable palm that generates changeable topology and augments dexterity and versatility of the hand. Structure design of the robotic hand is presented and based on mechanism decomposition kinematics of the metamorphic anthropomorphic hand is characterized with closed-form solutions leading to the workspace investigation of the robotic hand. With characteristic matrix equation, twisting motion of the metamorphic robotic hand is investigated to reveal both dexterity and manipulability of the metamorphic hand. Through a prototype, grasping and prehension of the robotic hand are tested to illustrate characteristics of the new metamorphic anthropomorphic hand.

Application of Dual-Number Matrices to the Inverse Kinematics Problem of Robot Manipulators

1985 ◽  
Vol 107 (2) ◽  
pp. 201-208 ◽  
Author(s):  
G. R. Pennock ◽  
A. T. Yang
Keyword(s):  
Closed Form ◽  
Inverse Kinematics ◽  
Robot Manipulators ◽  
Closed Form Solutions ◽  
Dual Number ◽  
Special Geometry ◽  
Inverse Kinematics Problem ◽  
Concise Representation ◽  
Joint Parameters ◽  
General Geometry

This paper presents the application of dual-number matrices to the formulation of displacement equations of robot manipulators with completely general geometry. Dual-number matrices make possible a concise representation of link proportions and joint parameters; together with the orthogonality properties of the matrices we are able to derive, in a systematic manner, closed-form solutions for the joint displacements of robot manipulators with special geometry as illustrated by three examples. It is hoped that the method presented here will provide a meaningful alternative to existing methods for formulating the inverse kinematics problem of robot manipulators.

Improving Elmore Model of RLC Networks for Applying to SWCNT Interconnects

2011 ◽  
Vol 110-116 ◽  
pp. 5078-5084
Author(s):  
Behrouz Behtoee ◽  
Rahim Faez
Keyword(s):  
Carbon Nanotube ◽  
Closed Form ◽  
Curve Fitting ◽  
Rise Time ◽  
Step Response ◽  
Analytical Estimate ◽  
Single Wall Carbon ◽  
Closed Form Solutions ◽  
Improved Accuracy ◽  
First Time

Elmore delay has been widely used as an analytical estimate of the interconnect delays in the performance-driven synthesis and layout of VLSI routing topologies. In this paper, Closed-form solutions for the 50% delay, rise time and overshoots of the step response of distributed Single Wall Carbon Nanotube (SWCNT), which consists RC and RLC parts, are presented for the first time. The proposed approach retains both efficiency and simplicity of the equivalent Elmore model with significantly improved accuracy, through surface fitting (3D) instead of curve fitting (2D).

Closed-Form Solutions for Eigenbuckling of Rectangular Mindlin Plate

2016 ◽  
Vol 16 (08) ◽  
pp. 1550079 ◽  
Author(s):  
Yufeng Xing ◽  
Wei Xiang
Keyword(s):  
Closed Form ◽  
Present Method ◽  
Separation Of Variables ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Major Development ◽  
First Time ◽  
Good Agreement

This paper studies the eigenbuckling of Mindlin plate with two adjacent edges clamped and the remaining edges simply supported or clamped by using the separation of variables method, and the concise and explicit closed-form solutions are obtained for the first time. The cases involving free edges can also be dealt with if there are two opposite edges simply supported. The closed-form solutions are in good agreement with the existing solutions, thus the validity of present method and accuracy of the obtained solutions are verified. This paper proves to be a major development of analytical method since it has long been acknowledged that the eigenbuckling of rectangular plates without two parallel edges simply supported are not amenable to analytical solutions.

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.

Closed Form Solutions of Joint Water-Filling for Coordinated Transmission

2010 ◽  
Vol E93-B (12) ◽  
pp. 3461-3468 ◽  
Author(s):  
Bing LUO ◽  
Qimei CUI ◽  
Hui WANG ◽  
Xiaofeng TAO ◽  
Ping ZHANG
Keyword(s):  
Closed Form ◽  
Closed Form Solutions ◽  
Water Filling
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