Robot Path Control with Al-Alaoui Rule for Fractional Calculus Discretization

Abstract

In this paper an application of the fractional calculus to path control is studied. The integer-order derivative and integral are replaced with the fractional-order ones in order to solve the inverse kinematics problem. As an approximation of the fractional differentiator the Al-Alaoui operator with power series expansion (PSE) is used. The proposed algorithm is a modification of the existing one based on Grunwald–Letnikov formula. In order to maintain the accuracy and to lower the memory requirements a history limit and a combination of fractional- and integer-order derivation are proposed. After reaching assumed accuracy or iteration limit the algorithm switches to integer order derivative and stops after few additional iterations. This approach allows to reduce the positional error and maintain the repeatability of fractional calculus approach. The simulated path in task space have been designed in a way that causes the instability of standard Closed Loop Pseudoinverse algorithm. Our study proves that use of fractional calculus may improve the joint paths.