Enhancing the performance in the offline controller tuning of robotic manipulators with chaos: a comparative study with differential evolution

The decision-making process regarding utilizing specific chaotic maps as number generators and the different ways they can be incorporated into differential evolution (DE), as well as the type of system and task for which chaos is functional in the controller tuning problem, is an open issue. This paper aims to address this need for more knowledge in the field by proposing sixteen different chaotic DE variants and presenting a formal study on the tuning of PID controllers through these variants for regulation and tracking tasks with robotic systems of incremental complexity. These systems include planar open-chain robotic manipulators with one, two, and six revolute joints. Concerning the proposed variants, they utilize pseudo-random numbers obtained from well-known chaotic maps, including Logistic, Sine, Hénon, and Lozi, at various DE stages such as initialization, mutation, crossover, and in all those stages. The comparative results based on descriptive and nonparametric statistics with respect to the proposed chaotic variants of DE and the DE using a general-purpose pseudo-random number generator such as Mersenne Twister (MT) indicates that using the single-state chaotic maps, especially the Sine map, increases the performance of the search process when used in the crossover stage or throughout the algorithm in complex systems. It is also observed that Sine map-based chaotic numbers can be useful enough during the search initialization for problems involving relatively less complex systems. Additionally, considering that chaotic maps require simple arithmetic operations, it is deduced that the variants that provide better results than the widely used MT also avoid the intricate operations involved in such a generator, reducing the computational burden to a good extent and increasing the area of application where this is a limitation.