By Ronald E. Mickens
This quantity offers a concise creation to the technique of nonstandard finite distinction (NSFD) schemes development and indicates how they are often utilized to the numerical integration of differential equations taking place within the traditional, 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 through different researchers. the significance of the e-book derives from its transparent and direct rationalization of NSFD within the introductory bankruptcy in addition to a huge dialogue of the long run instructions had to increase the subject.
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Additional info for Advances in the Applications of Nonstandard Finite Difference Schemes
2. Numerical Results The desired trajectories, space as follows: qid = qiI qid (i = 1 , 2 ) , are defined in the joint coordinate f (qiF - qiI)[lOt: - 1st: + 6tk] 43a) (43b) and q i F are the initial and final positions and, tI and t F are the initial and final time instants. Table 1 shows the values of these parameters. Figure 10 shows the desired trajectory of joint 1 and joint 2 during the task. qzI Table Joint 1 2 tl 3s 4s tF 5s 6s qi1 qiF -90" 0" -70" 10" The simulation study is first conducted using RK4 method.
K z l and ki2 are the control gains, and Gi(i=l,2)are defined in ( 3 5 ) and (36). Equation (11) is re-written in the following form: E = D [T - H(e + q,, Q + q d )- G ( e + q d ) ] - (38) qd where e = q - qd is the vector of joint coordinate errors. Alternatively, the above equation can be given as below: (39a) (39b) + + + + + where Ci = cos(ei q i d ) , Si = sin(ei q i d ) and Cij = cos(ei q i d ej q j d ) . In the state space form, Equations (39) can be written as follows (to be used Simulation Studies of Robotic Systems 31 by scheme I): 21 =22 (40) x3 = x 4 (15) (41) (12) R.
The maximum eigenvalues of these matrices are then calculated to determine the bound on effective step-size for stable numerical integration. 2. Numerical Results First, the dynamic equations of the manipulator are integrated using fourthorder Runge-Kutta (RK4) numerical scheme. OOls, which shows correct convergence. Figure 3 shows the simulation results using RK4 method but with larger step-sizes. For step-sizes greater than 0 . 0 5 4 ~ the ~ integration routine noticeably converges to wrong solutions.
Advances in the Applications of Nonstandard Finite Difference Schemes by Ronald E. Mickens