Inverse Kinematics of General 6R and 5R,P Serial Manipulators

1992 ◽  
Author(s):  
Dilip Kohli ◽  
Michael Osvatic
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Kinematics ◽  
Linear Equations ◽  
Computation Time ◽  
Solution Procedure ◽  
Linearization Method ◽  
Solution Technique ◽  
Serial Manipulators ◽  
General Geometry ◽  
Imaginary Roots

Abstract In this paper we present a solution to the inverse kinematics problem for serial manipulators of general geometry. The method is presented in detail as it applies to a 6R manipulator of general geometry. The equations used are derived using a linearization method and dialytic elimination. In doing this, all variables except one, a tangent half angle of a joint variable, can be eliminated. The result is a 16 by 16 matrix in which all terms are linear in the suppressed variable. The unique design of this matrix allows the suppressed variable to be solved as an eigenvalue problem. Substituting these values of the suppressed variable back into the equations, all other joint variables can be found using linear equations. The result is the 16 solutions expected for the 6R case. The same technique is also applicable to manipulators with prismatic joints. We present the solution technique for all six possible 5R,P manipulators through numerical examples. The primary distinction between the technique presented in this paper and recently published Raghavan and Roth (90a,b.c) solution is that they removed two known spurious imaginary roots of multiplicity four to obtain a 16th order polynomial for 6R and 5R,P cases. In our formulation, the 16th degree polynomial can be derived directly without having to remove any spurious imaginary roots. Another distinction is that the solution procedure presented in this paper can be reduced to an eigenvalue problem. This results in significant gains in computation time.

Inverse Kinematics of General 6R and 5R,P Serial Manipulators

1993 ◽  
Vol 115 (4) ◽  
pp. 922-931 ◽  
Author(s):  
D. Kohli ◽  
M. Osvatic
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Kinematics ◽  
Linear Equations ◽  
Computation Time ◽  
Solution Procedure ◽  
Solution Technique ◽  
Serial Manipulators ◽  
Order Polynomial ◽  
General Geometry ◽  
Imaginary Roots

In this paper we present a solution to the inverse kinematics problems for serial manipulators of general geometry. The method is presented in detail as it applies to a 6R manipulator of general geometry. The equations used are derived using power products and dialytic elimination. In doing this, all variables except one, a tangent half angle of a joint variable, can be eliminated. The result is a 16 by 16 matrix in which all terms are linear in the suppressed variable. The unique design of this matrix allows the suppressed variable to be solved as an eigenvalue problem. Substituting these values of the suppressed variable back into the equations, all other joint variables can be found using linear equations. The result is the 16 solutions expected for the 6R case. The same technique is also applicable to manipulators with prismatic joints. We present the solution technique for all six possible 5R,P manipulators through numerical examples. The primary distinction between the technique presented in this paper and the recently published Raghavan and Roth (1990) solution is that they removed two spurious imaginary roots of multiplicity four from a 24th order polynomial to obtain a 16th order polynomial for 6R and 5R,P cases. In our formulation, the 16th degree polynomial can be derived directly without having to remove any spurious imaginary roots. Another distinction is that the solution procedure can be reduced to an eigenvalue problem. This results in significant gains in computation time.

Inverse Kinematics of Serial-Chain Manipulators

1996 ◽  
Vol 118 (3) ◽  
pp. 396-404 ◽  
Author(s):  
Hong-You Lee ◽  
Charles F. Reinholtz
Keyword(s):  
Inverse Kinematics ◽  
Minimum Degree ◽  
Complete Solution ◽  
Linear Equations ◽  
End Effector ◽  
Serial Chain ◽  
Inverse Kinematics Problem ◽  
Univariate Polynomials ◽  
General Geometry

This paper proposes a unified method for the complete solution of the inverse kinematics problem of serial-chain manipulators. This method reduces the inverse kinematics problem for any 6 degree-of-freedom serial-chain manipulator to a single univariate polynomial of minimum degree from the fewest possible closure equations. It is shown that the univariate polynomials of 16th degree for the 6R, 5R-P and 4R-C manipulators with general geometry can be derived from 14, 10 and 6 closure equations, respectively, while the 8th and 4th degree polynomials for all the 4R-2P, 3R-P-C, 2R-2C, 3R-E and 3R-S manipulators can be derived from only 2 closure equations. All the remaining joint variables follow from linear equations once the roots of the univariate polynomials are found. This method works equally well for manipulators with special geometry. The minimal properties may provide a basis for a deeper understanding of manipulator geometry, and at the same time, facilitate the determination of all possible configurations of a manipulator with respect to a given end-effector position, the determination of the workspace and its subspaces with the different number of configurations, and the identification of singularity positions of the end-effector. This paper also clarifies the relationship between the three known solutions of the general 6R manipulator as originating from a single set of 14 equations by the first author.

Inverse Kinematics of General 4R2P, 3R3P, 4R1C, 2R2C, and 3C Serial Manipulators

1992 ◽  
Author(s):  
Dilip Kohli ◽  
Michael Osvatic
Keyword(s):  
Inverse Kinematics ◽  
Inverse Kinematic ◽  
Serial Manipulators ◽  
Linearly Independent ◽  
The Matrix ◽  
Half Angle ◽  
Inverse Kinematics Problem ◽  
Joint Variable ◽  
General Geometry ◽  
Solution Methodology

Abstract This paper presents a solution to the inverse kinematics problem for 4R2P, 3R3P, 4R1C, 2R2C and 3C manipulators of general geometry. The method used to solve these is based on a technique recently presented by the authors for solving the inverse kinematics of general 6R and 5R1P manipulators. In the 6R and 5R1P cases, the method initially starts using 14 linearly independent equations where as for the 4R2P, 3R3P, 4R1C, 2R2C and 3C manipulator only 3, 6, 7 or 10 linearly independent equations are required, depending on the case. Through the use of a linearization and dialytic elimination method all 4R2P, 3R3P, 4R1C, 2R2C and 3C cases are reduced to equating to zero the determinant of a matrix whose elements are linear in the tangent of a half angle of a joint variable. The size of this matrix is (8 × 8) for all 4R2P manipulators, (2 × 2) for all 3R3P and 3C manipulators, (16 × 16) for 4R1C manipulators, (4 × 4) for RCRC and CRCR manipulators and (8 × 8) for the remaining 2R2C manipulators providing 8th, 2nd, 16th, 4th and 8th degree inverse kinematic polynomial respectively. Thus, the determinant equated to zero gives us the characteristic equation of the degree expected. The unique form of the matrix allows us to obtain the solution by solving an eigenvalue problem. Many variations of the 4R2P, 3R3P, 4R1C, 2R2C and 3C manipulators are presented and the solution methodology is illustrated by several numerical examples.

A Novel Numerical Method to Solve the Inverse Kinematics of 3R Manipulators

2011 ◽  
Author(s):  
E. A. Gonza´lez-Barbosa ◽  
M. A. Gonza´lez-Palacios ◽  
L. A. Aguilera-Corte´s ◽  
C. A. Bernal-Marti´nez
Keyword(s):  
Differential Evolution ◽  
Numerical Method ◽  
Computational Efficiency ◽  
Software Package ◽  
Inverse Kinematics ◽  
Serial Manipulators ◽  
Singular Configurations ◽  
Simulation Results ◽  
General Geometry ◽  
Solution Problem

A new numerical method to solve the inverse kinematics solution problem of serial manipulators is developed in this paper. The proposed method is known as Differential Evolution (DE), a novel and efficient numerical method which has been adapted to solve the inverse kinematics solution of 3R serial manipulator of general geometry. Besides, the paper contains the complete structuring for the implementation of this new case in SnAP, a comprehensive software package for synthesis, analysis and simulation of serial manipulators. The DE method is stable since it converges to the solution with any initial values, and it is not sensitive to the singular configurations of serial manipulators. Simulation results are presented to show the performance benefits of the proposed algorithm. Computational efficiency of the method is shown based on the results, as well as in comparison with traditional methods used in this problem.

scholarly journals Research on the inverse kinematics of manipulator using an improved self-adaptive mutation differential evolution algorithm

2021 ◽  
Vol 18 (3) ◽  
pp. 172988142110144
Author(s):  
Qianqian Zhang ◽  
Daqing Wang ◽  
Lifu Gao
Keyword(s):  
Differential Evolution ◽  
Objective Function ◽  
Inverse Kinematics ◽  
Differential Evolution Algorithm ◽  
Weight Coefficient ◽  
Adaptive Mutation ◽  
Feasible Region ◽  
Serial Manipulators ◽  
De Algorithm ◽  
Self Adaptive

To assess the inverse kinematics (IK) of multiple degree-of-freedom (DOF) serial manipulators, this article proposes a method for solving the IK of manipulators using an improved self-adaptive mutation differential evolution (DE) algorithm. First, based on the self-adaptive DE algorithm, a new adaptive mutation operator and adaptive scaling factor are proposed to change the control parameters and differential strategy of the DE algorithm. Then, an error-related weight coefficient of the objective function is proposed to balance the weight of the position error and orientation error in the objective function. Finally, the proposed method is verified by the benchmark function, the 6-DOF and 7-DOF serial manipulator model. Experimental results show that the improvement of the algorithm and improved objective function can significantly improve the accuracy of the IK. For the specified points and random points in the feasible region, the proportion of accuracy meeting the specified requirements is increased by 22.5% and 28.7%, respectively.

A Method to Design Shell-Side Pressure Drop Constrained Tubular Heat Exchangers

1977 ◽  
Vol 99 (3) ◽  
pp. 441-448 ◽  
Author(s):  
K. P. Singh ◽  
M. Holtz
Keyword(s):  
Pressure Drop ◽  
Heat Exchangers ◽  
Solution Procedure ◽  
Design Guidelines ◽  
Solution Technique ◽  
Nuclear Heat ◽  
Side Pressure ◽  
Flow Induced Vibrations ◽  
The Subject ◽  
Shell And Tube

In shell and tube heat exchangers, the triple segmental baffle arrangement has been infrequently used, even though the potential of this baffle system for high thermal effectiveness with low pressure drop is generally known. This neglect seems to stem from the lack of published design guidelines on the subject. Lately, however, with the rapid growth in the size of nuclear heat exchangers, the need to develop unconventional baffling pattern has become increasingly important. A method to effectively utilize the triple segmental concept to develop economical designs is presented herein. The solution technique given in this paper is based on a flow model named “Piecewise Continuous Cosine Model.” The solution procedure easily lends itself to detailed analysis to determine safety against flow-induced vibrations.

An efficient approach to the seismogram synthesis for a basin structure using propagation invariants

1996 ◽  
Vol 86 (2) ◽  
pp. 379-388 ◽  
Author(s):  
H. Takenaka ◽  
M. Ohori ◽  
K. Koketsu ◽  
B. L. N. Kennett
Keyword(s):  
Integral Equation ◽  
Inverse Fourier Transform ◽  
Linear Equations ◽  
Computation Time ◽  
Reduction Technique ◽  
Final Solution ◽  
Space Domain ◽  
New Approach ◽  
Simultaneous Linear Equations ◽  
Single Integral

Abstract The Aki-Larner method is one of the cheapest methods for synthetic seismograms in irregularly layered media. In this article, we propose a new approach for a two-dimensional SH problem, solved originally by Aki and Larner (1970). This new approach is not only based on the Rayleigh ansatz used in the original Aki-Larner method but also uses further information on wave fields, i.e., the propagation invariants. We reduce two coupled integral equations formulated in the original Aki-Larner method to a single integral equation. Applying the trapezoidal rule for numerical integration and collocation matching, this integral equation is discretized to yield a set of simultaneous linear equations. Throughout the derivation of these linear equations, we do not assume the periodicity of the interface, unlike the original Aki-Larner method. But the final solution in the space domain implicitly includes it due to use of the same discretization of the horizontal wavenumber as the discrete wavenumber technique for the inverse Fourier transform from the wavenumber domain to the space domain. The scheme presented in this article is more efficient than the original Aki-Larner method. The computation time and memory required for our scheme are nearly half and one-fourth of those for the original Aki-Larner method. We demonstrate that the band-reduction technique, approximation by considering only coupling between nearby wavenumbers, can accelerate the efficiency of our scheme, although it may degrade the accuracy.

Numerical Linear Algebra

1966 ◽  
Vol 9 (05) ◽  
pp. 757-801 ◽  
Author(s):  
W. Kahan
Keyword(s):  
Eigenvalue Problem ◽  
Linear Algebra ◽  
Linear Equations ◽  
Numerical Linear Algebra ◽  
System Of Linear Equations ◽  
Image Position

The primordial problems of linear algebra are the solution of a system of linear equations and the solution of the eigenvalue problem for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

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