Trackability analysis and tracking control of swirling pendulum -an underactuated mechanical system

Underactuated mechanical systems are essentially the systems which have lesser number of free control actuators than degrees of freedom. Due to their diverse applications and potential usage in multiple disciplines, tracking and control of this particular class of system has become an important research problem in the _eld of dynamics and control. There is a vast body of research on controlling this class of systems to enable such system to exploit its inherent dynamics in order to become agile, robust and e_cient. But, there is a limited literature available on solving a larger and important question on ability of a system to track any particular reference command. With this work, we have done preliminary analysis in the direction of answering this question by analyzing the 'trackability' of a system. The analysis is done by means of developing a new system which we refer to it as 'swirling pendulum' and an experimental setup of the same has been built from scratch as part of this work. Although many physical systems have been studied in control literature, this system possesses richer dynamics and deep coupling among its degrees of freedom. The Swirling pendulum has essentially two links, one of which swings in one plane and the other one swirls in di_erent planes. Due to its coupled dynamics, we see interesting behavior of this system about its static equilibria. Simpli_ed mathematical model of swirling pendulum is developed. This model is then improved by introducing non conservative forces like friction and motor dynamics whose parameters are identi_ed using standard system identi_cation techniques. We then quantify the ability to track the reference commands by analyzing the trackability index of di_erent reference commands using Markov parameter matrix of system. Also, projections of these reference commands on markov parameter reveal the closeness with which a particular trajectory can be partially tracked if not tracked perfectly. We validate these results with simulation on nonlinear plant model as well as experimentally comparing the tracking performance.