Control of an underactuated system around a periodic orbit

Duyul, Ayşe Deniz
Quasi-periodic behavior is one of the most important fundamental building blocks for locomotion in biological (and robotic) systems. The dynamics that govern the motion of such behaviors are generally highly nonlinear and underactuated. One method of analyzing the quasi-periodic behaviors of such systems is to linearize the system around these periodic trajectories. Such a linearization provides us a linear time periodic (LTP) system around the neighborhood of the periodic orbit. Analysis and control of LTP systems has gained some attention in the last decade and various methods have emerged for this purpose. One of these methods, which we focus on this thesis, is the harmonic balance approach. In the literature harmonic balance and harmonic transfer functions based approaches have been mostly utilized in system identification. There are very few methods that adopt these approaches for control and stabilization purposes. In this thesis, we propose a control method which can generate stable periodic behaviors for underactuated nonlinear robotic systems. We first generate a reference periodic orbit that can be tracked by the given system and find the open-loop control input that can achieve this trajectory. We then linearize the equations of motion to obtain an LTP representation around the given periodic orbit. Finally, we propose a linear time invariant state feedback control law based on eigenvalue optimization of the lifted LTI representation of the LTP system (harmonic balance). We applied our method on the widely used cart-pendulum example in simulation environment for limit cycles around both stable and unstable equilibrium points. We obtained succesful results for limit cycles around the stable equilibrium point enhancing system performance and tracking position, amlitude and phase changes in the limit cycle. For limit cycles around the unstable equilibrium point, we obtained stable periodic orbits and promising directions for future studies.
Citation Formats
A. D. Duyul, “Control of an underactuated system around a periodic orbit,” M.S. - Master of Science, Middle East Technical University, 2018.