Fourier series-based algorithm for control optimization in pendulum capsule drive: an integrated computational and experimental study

This work addresses the control of non-smooth and discontinuous mechanical systems, exemplified by pendulum capsule drives. The previously developed Fourier series-based control optimization framework is extended by incorporating a greedy continuation strategy, in which the optimization horizon and the number of Fourier harmonics are progressively increased while preserving the quality of previously obtained solutions. The proposed approach is intended for nonlinear systems, including those exhibiting non-smooth or discontinuous behavior due to phenomena such as dry friction and impacts. A detailed comparison of the numerical results obtained with the non-greedy and greedy continuation strategies is presented and discussed. Experimental validation using a prototype device confirms the method’s effectiveness, with results closely matching numerical simulations. The system achieves average locomotion speeds of 2.48 cm/s and 2.58 cm/s for optimized pendulum angular velocity profiles composed of three and six Fourier harmonics, respectively. In numerical simulations, a possible further performance gain with twelve harmonics was observed.