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

Sliding mode control of the reaction wheel pendulum

textThe Reaction Wheel Pendulum (RWP) is an interesting nonlinear system. A prototypical control problem for the RWP is to stabilize it around the upright position starting from the bottom, which is generally divided into at least 2 phases: (1) Swing-up phase: where the pendulum is swung up and moves toward the upright position. (2) Stabilization phase: here, the pendulum is controlled to be balanced around the upright position. Previous studies mainly focused on an energy method in swing-up phase and a linearization method in stabilization phase. However, several limitations exist. The energy method in swing-up mode usually takes a long time to approach the upright position. Moreover, its trajectory is not controlled which prevents further extensions. The linearization method in the stabilization phase, can only work for a very small range of angles around the equilibrium point, limiting its applicability. In this thesis, we took the 2nd order state space model and solved it for a constant torque input generating the family of phase-plane trajectories (see Appendix A). Therefore, we are able to plan the motion of the reaction wheel pendulum in the phase plane and a sliding mode controller may be implemented around these trajectories. The control strategy presented here is divided into three phases. (1) In the swing up phase a switching torque controller is designed to oscillate the pendulum until the system’s energy is enough to drive the system to the upright position. Our approach is more generic than previous approaches; (2) In the catching phase a sliding surface is designed in the phase plane based on the zero torque trajectories, and a 2nd order sliding mode controller is implemented to drive the pendulum moving along the sliding surface, which improves the robustness compared to the previous method in which the controller switches to stabilization mode when it reaches a pre-defined region. (3) In the stabilization phase a 2nd order sliding mode integral controller is used to solve the balancing problem, which has the potential to stabilize the pendulum in a larger angular region when compared to the previous linearization methods. At last we combine the 3 phases together in a combined strategy. Both simulation results and experimental results are shown. The control unit is National Instruments CompactRIO 9014 with NI 9505 module for module driving and NI 9411 module for encoding. The Reaction Wheel Pendulum is built by Quanser Consulting Inc. and placed in UT’s Advanced Mechatronics Lab.Mechanical Engineerin

Controlled Lagrangian Methods and Tracking of Accelerated Motions

Matching techniques are applied to the problem of stabilization of uniformly accelerated motions of mechanical systems with symmetry. The theory is illustrated with a simple model-a wheel and pendulum system