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# A Spherical Pendulum System To Teach Key Concepts in Kinematics, Dynamics, Control, and Simulation

Fathi  Ghorbel, Member, IEEE, and James B. Dabney

Abstract -

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 (un-actuated) 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, we illustrate each step in the process of kinematic and dynamic modeling, simulation, and control of the SPENDULAP. We 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, we illustrate the development of a numerical simulation of the SPENDULAP dynamics, and provide examples of uses of simulation and animation using MATLAB. Finally, we show the development of linear and nonlinear control laws, and illustrate testing them using simulation.

### Author Contact Information

Fathi Ghorbel
Department of Mechanical Engineering
Rice University
6100 Main Street - MS 321
Houston, TX 77005
Phone: (713) 527-8101 ext. 3738
Fax: (713) 285-5423
E-mail: ghorbel@rice.edu

James B. Dabney
Department of Mechanical Engineering
Rice University
6100 Main Street - MS 321
Houston, TX 77005
Phone: (713) 527-8101 ext. 3619
Fax: (713) 285-5423
E-mail: jdabney@rice.edu

### Author Biographies

Fathi Ghorbel is an Associate Professor at the Department of Mechanical Engineering and the Department of Bioengineering at Rice University, Houston, Texas where he is also the Director of the Dynamic Systems and Control Laboratory, and the Robotics Laboratory, and the Co-Director of the Biomedical Systems and Instrumentation Lab. His research is in the areas of systems and control theory, robotics, and biomedical engineering systems. Dr. Ghorbel received the B.S. degree with honors from the Pennsylvania State University in 1985, the M.S. degree from Carnegie-Mellon University in 1987, and the Ph.D. degree from the University of Illinois at Urbana-Champaign in 1991 all in Mechanical Engineering. Dr. Ghorbel is a member of Sigma Xi, IEEE, ASME, IFAC, SIAM, and IASTED. He is an Associate Editor for the IEEE Transactions on Control Systems Technology, an Associate Editor for the International Journal of Robotics and Automation, and an Associate Editor for the IEEE Control Systems Society Conference Editorial Board.

James B. Dabney is a Lecturer and Post-Doctoral Fellow in the Department of Mechanical Engineering at Rice University, Houston, Texas. His research is in the application of numerical optimization and optimal control techniques to problems in mechanical engineering, numerical simulation, and control. Dr. Dabney received the B.S. degree in Mechanical Engineering from Virginia Polytechnic Institute and State University 1974, the M.S. degree in Process Monitoring and Control from University of Houston - Clear Lake in 1993, and the Ph.D. in Mechanical Engineering from Rice University in 1998. Dr. Dabney is a member of AIAA and ASME and a reviewer for the International Journal of Robotics and Automation, Simulation, and IEEE Transactions on Control Systems Technology.