A spherical pendulum system to teach key concepts in kinematics, dynamics, control, and simulation

The Rice Spherical PENDUlum Laboratory APparatus (SPENDULAP) is a rich teaching aid for senior and first year graduate courses in kinematics, dynamics, control and simulation. It consists of a free-swinging (unactuated) rigid pendulum mounted in a rotating frame. The frame rotates about an axis perpendicular to the pendulum swing axis and is driven by an electric motor. The SPENDULAP is attractive as a teaching tool because it is easy to visualize the motion of the pendulum, but somewhat challenging to model kinematically and dynamically. In particular, the three-dimensional nature of the pendulum motion allows students to gain proficiency and confidence not possible with a planar apparatus. Additionally, nonlinearities in the dynamics present interesting, but tenable, control challenges. In this paper, the authors illustrate each step in the process of kinematic and dynamic modeling, simulation and control of the SPENDULAP. They start with the kinematic analysis and then develop the equations of motion using both Newtonian and Lagrangian approaches. The spherical pendulum is sufficiently complex to demonstrate the advantages of the Lagrangian approach, and it also offers an excellent illustration of the benefits of the Newtonian formulation. Next, they illustrate the development of a numerical simulation of the SPENDULAP dynamics, and provide examples of uses of simulation and animation using MATLAB. Finally, they show the development of linear and nonlinear control laws, and illustrate testing them using simulation