Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Homotopy Continuation

2002 ◽  
Vol 124 (4) ◽  
pp. 652-661 ◽  
Author(s):  
Eric Lee ◽  
Constantinos Mavroidis
Keyword(s):  
Design Problem ◽  
Free Choice ◽  
Robot Manipulators ◽  
Continuation Method ◽  
Geometric Design ◽  
Design Parameters ◽  
Homotopy Continuation ◽  
Serial Link ◽  
End Effector ◽  
Homotopy Continuation Method

In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints is solved using a polynomial homotopy continuation method. Three spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3-R manipulator are computed so that the manipulator will be able to place its end-effector at these three pre-specified locations. Denavit and Hartenberg parameters and 4×4 homogeneous matrices are used to formulate the problem and obtain eighteen design equations in twenty-four design unknowns. Six of the design parameters are set as free choices and their values are selected arbitrarily. Two different cases for selecting the free choices are considered and their design equations are solved using polynomial homotopy continuation. In both cases for free choice selection, eight distinct manipulators are found that will be able to place their end-effector at the three specified spatial positions and orientations.

An Elimination Procedure for Solving the Geometric Design of Spatial 3R Manipulators

2005 ◽  
Vol 128 (1) ◽  
pp. 142-145 ◽  
Author(s):  
Eric Lee ◽  
Constantinos Mavroidis
Keyword(s):  
Design Problem ◽  
Geometric Design ◽  
Geometric Parameters ◽  
The Other ◽  
Design Parameters ◽  
Single Variable ◽  
Serial Link ◽  
End Effector ◽  
Elimination Procedure ◽  
First Time

In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints when three precision points are specified is solved using an algebraic elimination method for the first time. Three spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3R manipulator are computed so that the manipulator will be able to place its end-effector at these three prespecified locations. In this problem, six of the design parameters are set as free choices and their values are selected arbitrarily. For the specific case studied in this paper, a 12 deg single variable polynomial is calculated that has eight roots that are the design solutions and the other four roots are extraneous solutions.

A New Polynomial Solution to the Geometric Design Problem of Spatial R-R Robot Manipulators Using the Denavit and Hartenberg Parameters

1999 ◽  
Vol 123 (1) ◽  
pp. 58-67 ◽  
Author(s):  
Constantinos Mavroidis ◽  
Eric Lee ◽  
Munshi Alam
Keyword(s):  
Design Problem ◽  
Degrees Of Freedom ◽  
Robot Manipulators ◽  
Polynomial Solution ◽  
Open Loop ◽  
Geometric Design ◽  
New Method ◽  
Design Parameters ◽  
End Effector ◽  
Spatial Locations

This paper presents a new method to solve the geometric design problem of spatial two degrees of freedom, open loop robot manipulators with revolute joints that perform tasks, which require the positioning of the end-effector in three spatial locations. Tsai and Roth [3] solved this problem first using screw parameters to describe the kinematic topology of the R-R manipulator and screw displacements to obtain the design equations. The new method, which is developed in this paper, uses Denavit and Hartenberg parameters and 4×4 homogeneous matrices to formulate and obtain the kinematic equations. The loop-closure geometric equations provide eighteen design equations in eighteen unknowns. Polynomial Elimination techniques are used to solve these equations and obtain the manipulator Denavit and Hartenberg parameters and the manipulator base and end-effector geometric parameters. A sixth order polynomial is obtained in one of the design parameters. Only two of the six roots of the polynomial are real and they correspond to two different robot manipulators that can reach the desired end-effector poses.

An Algebraic Elimination Based Algorithm for Solving the Geometric Design Problem of Spatial 3R Manipulators

2004 ◽  
Author(s):  
Eric Lee ◽  
Constantinos Mavroidis
Keyword(s):  
Design Problem ◽  
Geometric Design ◽  
Geometric Parameters ◽  
The Other ◽  
Design Parameters ◽  
Single Variable ◽  
Serial Link ◽  
Elimination Method ◽  
End Effector ◽  
First Time

In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints when three precision points are specified is solved using an algebraic elimination method for the first time. Three spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3-R manipulator are computed so that the manipulator will be able to place its end-effector at these three pre-specified locations. In this problem, six of the design parameters are set as free choices and their values are selected arbitrarily. For the specific case studied in this paper, a twelve-degree single variable polynomial is calculated that has eight roots that are the design solutions and the other four roots are extraneous solutions.

Analytic Geometric Design of Spatial R-R Robot Manipulators

2000 ◽  
Author(s):  
Constantinos Mavroidis ◽  
Munshi Alam ◽  
Eric Lee
Keyword(s):  
Degrees Of Freedom ◽  
Robot Manipulators ◽  
Open Loop ◽  
Geometric Design ◽  
Sixth Order ◽  
Design Parameters ◽  
End Effector ◽  
Revolute Joints ◽  
Order Polynomial ◽  
Spatial Locations

Abstract This paper studies the geometric design of spatial two degrees of freedom, open loop robot manipulators with revolute joints that perform tasks, which require the positioning of the end-effector in three spatial locations. This research is important in situations where a robotic manipulator or mechanism with a small number of joint degrees of freedom is designed to perform higher degree of freedom end-effector tasks. The loop-closure geometric equations provide eighteen design equations in eighteen unknowns. Polynomial Elimination techniques are used to solve these equations and obtain the manipulator Denavit and Hartenberg parameters. A sixth order polynomial is obtained in one of the design parameters. Only two of the six roots of the polynomial are real and they correspond to two different robot manipulators that can reach the desired end-effector poses.

Five Precision Point Synthesis of Spatial RRR Manipulators Using Interval Analysis

2004 ◽  
Vol 126 (5) ◽  
pp. 842-849 ◽  
Author(s):  
Eric Lee ◽  
Constantinos Mavroidis ◽  
Jean Pierre Merlet
Keyword(s):  
Interval Analysis ◽  
Design Problem ◽  
Geometric Design ◽  
Geometric Parameters ◽  
Analysis Method ◽  
Serial Link ◽  
End Effector ◽  
Interval Method ◽  
Interval Analysis Method ◽  
First Time

In this paper, the geometric design problem of serial-link robot manipulators with three revolute (R) joints is solved for the first time using an interval analysis method. In this problem, five spatial positions and orientations are defined and the dimensions of the geometric parameters of the 3-R manipulator are computed so that the manipulator will be able to place its end-effector at these pre-specified locations. Denavit and Hartenberg parameters and 4×4 homogeneous matrices are used to formulate the problem and obtain the design equations and an interval method is used to search for design solutions within a predetermined domain.

scholarly journals New collocation path-following approach for the optimal shape parameter using Kernel method

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Zouhair Saffah ◽  
Abdelaziz Timesli ◽  
Hassane Lahmam ◽  
Abderrahim Azouani ◽  
Mohamed Amdi
Keyword(s):  
Shape Parameter ◽  
Kernel Method ◽  
Continuation Method ◽  
Path Following ◽  
High Order ◽  
Homotopy Continuation ◽  
Homotopy Continuation Method ◽  
Optimal Value ◽  
Rbf Kernel ◽  
Order Algorithm

AbstractThe goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given.

Pose Accuracy Analysis of Robot Manipulators Based on Kinematics

2011 ◽  
Vol 201-203 ◽  
pp. 1867-1872 ◽  
Author(s):  
Jian Ye Zhang ◽  
Chen Zhao ◽  
Da Wei Zhang
Keyword(s):  
Matrix Theory ◽  
Robot Manipulators ◽  
Error Model ◽  
Robot Kinematics ◽  
Accuracy Analysis ◽  
Serial Link ◽  
End Effector ◽  
Surface Graph ◽  
Position And Orientation ◽  
Kinematics Parameters

The pose accuracy of robot manipulators has long become a major issue to be considered in its advanced application. An efficient methodology to generate the end-effector position and orientation error model of robotic manipulator has been proposed based on the differential transformation matrix theory. According to this methodology, a linear error model that described the end-effector position and orientation errors due to robot kinematics parameters errors has been presented. A computer program to generate the error model and perform the accuracy analysis on any serial link manipulator has been developed in MATLAB. This methodology and software are applied to the accuracy analysis of a Phantom Desktop manipulator. The positioning error of the manipulator in its workspace cross section (XOZ) has been plotted as 3D surface graph and discussed.

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