Application of nonstandard finite difference schemes to the simulation studies of robotic systems

The application of the nonstandard finite difference schemes to the challenging task of obtaining stable numerical solutions for highly nonlinear and coupled differential equations that describe the dynamics of robotic manipulators is investigated in this chapter. It is shown that despite its simplicity, an appropriate form of discrete derivatives of the original differential equations could greatly reduce numerical instabilities while expediting computation time. Here, two nonstandard schemes are employed to construct the discrete derivatives. In the first scheme, the orders of the discrete derivatives are equal to the orders of the corresponding derivatives of the differential equations and, the denominators of the discrete derivatives take on a more complicated function of the step-size to ensure that the fixed (equilibrium) points of the resulting discrete system has the same stability properties as those of the original system. The second scheme has the same characteristics as the first one, with the addition of having the nonlinear terms replaced by nonlccal discrete representations. Both schemes are evaluated and compared with the popular fourth-order Runge-Kutta method, through simulating the motion of a two degree-of-freedom planar manipulator. It is demonstrated that firstly, using nonstandard schemes, the possibility of having spurious solutions is eliminated. Secondly, nonstandard finite difference derivatives are numerically stable given any large step-sizes. This is significant since using nonstandard schemes, numerical simulation of the complex dynamics of robotic systems can potentially be expedited, thereby allowing close to real-time simulations of robotic systems with reliable results.