A Spherical Pendulum Modeling & Control

The goal of this semester project was to set up the mathematical model for the simplified process of the spherical pendulum, develop a simulation model of the process in Modelica that could then also be animated in 3D, and finally, design a controller for the process. The simplification is that the pendulum cannot turn about its own axis. For the Modelica model, a Pendulum Library, with the essential parts, was created that could then just be selected and connected to assemble the desired model. Additionally, the parts from the Modelica MultiBody Library were simplified so that the user does not have to calculate various vectors and inertia tensors. With this Pendulum Library, a two, three and four wheel variation of the spherical pendulum was built, of which finally the three wheel model was of greatest interest. To verify the correctness of the Modelica model, it was linearized and its state-space matrices were compared to the ones resulting from linearizing the equivalent mathematical model. The comparison yielded only slight deviations between the two models, allowing the conclusion that the Modelica model is physically correct. For the controller, a state-feedback controller was implemented. Using Matlab, the L vector for the controller was calculated. A variety of different L vectors corresponding to different pole placements as well as various reference signals were tested. The results were as follows. To achieve the most ideal reference tracking with the least error and actuator effort, The poles should have the same frequency as the natural frequency of the spherical pendulum. Increasing the damping of the poles beyond 45DA decreases the error minimally. The frequency of the reference trajectory to track should also be equal to the natural frequency of the spherical pendulum. The amplitude of the reference trajectory should not be too large since the model has been linearized around the stable equilibrium of the spherical pendulum