Discrete Mathematics

Advances in the Applications of Nonstandard Finite by Ronald E. Mickens PDF

By Ronald E. Mickens

ISBN-10: 9812564047

ISBN-13: 9789812564047

ISBN-10: 9812703314

ISBN-13: 9789812703316

This quantity offers a concise advent to the technique of nonstandard finite distinction (NSFD) schemes building and exhibits how they are often utilized to the numerical integration of differential equations happening within the normal, biomedical, and engineering sciences. those tools had their genesis within the paintings of Mickens within the 1990's and at the moment are commencing to be greatly studied and utilized via different researchers. the significance of the publication derives from its transparent and direct clarification of NSFD within the introductory bankruptcy in addition to a large dialogue of the longer term instructions had to improve the subject.

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Simulation results using nonstandard scheme I1 with different effective stepsizes. Simulation Studies Fig. 9. of Robotic Sgstems 29 Simulation results using nonstandard scheme I1 with large effective step-sizes. 30 R. F. Abo-Shanab, N . Sepehri and C. Q. 2. 1. Description of the Case In this case study, the manipulator is programmed to follow a desired trajectory using a proportional-derivative controller with gravity compensation. Thus, control torques IT^ (i = 1 , 2 ) are defined as follows: (37) where ei = (4%- q 2 d ) and 6%= (qi - q , d ) are the error and error derivative a t the manipulator joints, respectively.

20. Comparison between simulation results of tracking errors using small and large values for the step-size h (scheme 11). 44 R. F. Abo-Shanab, N . Sepehri and C. Q. Wu Time (s) Fig. 21. 001s and h = 10s). 5 . Conclusions In this chapter, two nonstandard finite difference (NSFD) schemes were employed to, for the first time, study numerical solutions of the equations of motion of a typical robot manipulator. The simulation results were also compared with the ones obtained using the conventional fourth-order Runge-Kutta (RK4) algorithm.

Rewriting Equation (13), we have 1% (20) where x = {XI,. . ,22,.. , x n }T and f = { f l ,f2,. . , f , } T . Knowing that at fixed (equilibrium) point, R = (21, 2 2 , . . , Z n } T , f ( R ) = 0 , one can write (21) Simulation Studies of Robotic Systems 17 where ek = (Xk- %) = { E ! , $, . . ,E;}' represents any perturbation about the fixed point. Now, consider the value of the parameter A' that makes the scheme numerically stable given any value of the parameter h. From Equations ( 2 0 ) and (21), we have, -@ l-e-A*h A' af =-( dxk E x k (22) Assume that function f is sufficiently smooth, then the Jacobian matrix R = PIx.

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Advances in the Applications of Nonstandard Finite Difference Schemes by Ronald E. Mickens

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