Analysis and Control of Underactuated Mechanical Systems by Energy Shaping Methods

This diploma thesis mainly focuses on the analysis and control of underactuated mechanical systems by control methods which modify the closed loop’s total energy. On the one hand the Euler-Lagrange and on the other hand the Hamiltonian picture are very useful in this context (chapter 2) which lead to the Method of Controlled Lagrangians or rather Interconnection and Damping Assignment - Passivity Based Control IDA-PBC). In the case of underactuated systems these methods lead to a set of non-linear PDEs.Solving these PDEs can be quite a complex issue for concrete examples. Therefore, a special class of systems is introduced: on the one hand this class allows to fulfil these PDEs under certain conditions on the desired closed-loop’s total energy for the Method of Controlled Lagrangians (chapter 3) and on the other hand to reduce these PDEs to a set of non-linear ODEs in the case of IDA-PBC (chapter 4) which can be managed more easily. The well-known underactuated cart and pendulum model, for instance, corresponds to the special class of systems. Thereby, it is not possible to globally swing up the pendulum with these methods. However, quite a large region of attraction can be obtained for stabilising the upright equilibrium of the pendulum with a desired displacement of the cart. Finally, an approach which allows to swing up the pendulum where the physical energy of the system is forced to a desired energy level by feedback, is presented (chapter 5). Furthermore, this approach and the former introduced methods are merged to create a hybride system which is able to swing up the pendulum and to stabilise the upright equilibrium of the pendulum with a desired cart displacement. Another objective of this thesis is to test these control methods on the laboratory setup cart and pendulum and to compare simulation results and measurement.