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Purpose: The aim of this article is focused on providing numerical solutions for system of second order robot arm problem using the Runge-Kutta Sixth order algorithm. Design/methodology/approach: The parameters governing the arm model of a robot control problem have also been discussed through RK-sixth-order algorithm. The precised solution of the system of equations representing the arm model of a robot has been compared with the corresponding approximate solutions at different time intervals. Findings: Results and comparison show the efficiency of the numerical integration algorithm based on the absolute error between the exact and approximate solutions. The stability polynomial for the test equation γ=λγ (�γ is a complex Number) using RK-butcher algorithm obtained by Murugesan et. al. [1] and Park et. al. [2,3] is not correct and the stability regions for RK-Butcher methods have been absurdly presented. They have made a blunder in determining the range for real parts of �λh (h is a step size) involved in the test equation for RK-Butcher algorithms. Further, they have abruptly drawn the stability region for STWS method assuming that it is based on the Taylor's series technique. Research limitations/implications: It is noticed that STWS algorithm is not based on the Taylor�'s series method and it is an A-stable method. In the present paper, a corrective measure has been taken to obtain the stability polynomial for the case of RK-Butcher algorithm, the ranges for the real part of �λh and to present graphically the stability regions of the RK-Butcher methods. Originality/value: Based on the numerical results and graphs, a thorough comparison is carried out between the numerical algorithms.
Wydawca
Rocznik
Tom
Strony
38--44
Opis fizyczny
Bibliogr. 21 poz., tab., wykr.
Twórcy
autor
autor
- Department of Mathematics, National Institute of Technology, Tiruchirappalli-620 015, Tamilnadu, India, rpalagu@nitt.edu
Bibliografia
- [1] K. Murugesan, N.P. Gopalan, D. Gopal, Error free butcher algorithms for linear electrical circuits, ETRI Journal 27/2 (2005) 195-205.
- [2] J.Y. Park, D.J. Evans, K. Murugesan, S. Sekar, V. Murugesh, Optimal Control of Singular Systems using the RK-Butcher Algorithm, International Journal of Computer Mathematics 81/2 (2004) 239-249.
- [3] J.Y. Park, K. Murugesan, D.J. Evans, S. Sekar, V. Murugesh, Observer Design of Singular Systems (transistor circuits) using the RK-Butcher Algorithm, International Journal of Computer Mathematics 82/1 (2004) 111-123.
- [4] S. Sekar, V. Murugesh, K. Murugesan, Numerical Strategies for the System of Second order IVPs Using the RK-Butcher Algorithms, International Journal of Computer Science and Applications 1/2 (2004) 96-117.
- [5] Z. Taha, Approach to Variable Structure control of Industrial Robots, in: Robot Control-theory and Applications, Peter Peregrinus Ltd, North-Holland, 1988, 53-59.
- [6] S. Oucheriah, Robust tracking and model following of uncertain dynamic delay systems by memory less linear controllers, IEEE Transactions on Automatic Control 44/7 (1999) 1473-1481.
- [7] D. Lim, H. Seraji, Configuration control of a mobile dexterous robot: real time implementation and experimentation, International Journal of Robotics Research 16/5 (1997) 601-618.
- [8] M.M. Polvcarpou, P.A. Loannou, A Robust adaptive non-linear control desig, Automatica 32/3 (1996) 423-427.
- [9] H. Krishnan, H.N. Mcclamroch, Tracking in non-linear differential- algebraic control systems with applications to constrained robot systems, Automatica 30/12 (1994) 1885-1897.
- [10] Q. Zhihua, Robot Control of a class of non-linear uncertain systems, IEEE Transactions on Automatic Control 37/9 (1992) 1437-1442.
- [11] R.K. Alexander, J.J. Coyle, Runge-Kutta methods for differential- algebric systems, SIAM Journal of Numerical Analysis 27/3 (1990) 736-752.
- [12] D.J. Evans, A new 4th Order Runge-Kutta method for initial value problems with error control, International Journal of Computer Mathematics 139 (1991) 217-227.
- [13] C. Hung, Dissipativity of Runge-Kutta methods for dynamical systems with delays, IMA Journal of Numerical Analysis 20 (2000) 153-166.
- [14] L.F. Shampine, H.A. Watts, The art of a Runge-Kutta code. Part I, Mathematical Software 3 (1977) 257-275.
- [15] J.C. Butcher, On Runge proccesses of higher order, Journal of Australian Mathematical Society 4 (1964) 179.
- [16] J.C. Butcher, The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods, John Wiley and Sons, U.K., 1987.
- [17] J.C. Butcher, On order reduction for Runge-Kutta methods applied to differential-algebraic systems and to stiff systems of ODEs, SIAM Journal of Numerical Analysis 27 (1990) 447-456.
- [18] L.F. Shampine, M.K. Gordon Computer solutions of ordinary differential equations, W.H. Freeman, San Francisco, 1975.
- [19] D. Gopal, V. Murugesh, K. Murugesan, Numerical solution of second-order robot arm control problem using Runge-Kutta-Butcher algorithm, International Journal of Computer Mathematics 83/3 (2006) 345-356.
- [20] Z. Taha, Dynamics and Control of Robots, Ph.D Thesis, University of Wales, 1987.
- [21] H.P. Huang W.L. Tseng, Asymtotic observer design for constrained robot systems, IEE Proceedings D: Control Theory and Applications 138/3 (1991) 211-216.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA9-0042-0006