Introduction

A major challenge facing instructors teaching advanced undergraduate courses in kinematics, dynamics, control, and simulation is helping students relate the theory to the physical world. Our goal in selecting a laboratory apparatus for this purpose was to identify a physical system that would be sufficiently complex to challenge the students, but not overwhelming. The Rice Spherical Pendulum Laboratory Apparatus (SPENDULAP) (Figure 1) has proven to be an excellent teaching aid for this purpose. In particular, the three-dimensional nature of the pendulum motion allows students to gain proficiency and confidence not possible with a planar apparatus.

We have used the SPENDULAP in an advanced undergraduate and introductory graduate dynamics and control course at Rice University for a number of years. The initial project was an expansion on a textbook problem from [1]. Design and fabrication of the SPENDULAP was a Senior Engineering Design Project for a four-student team in the 1996-1997 academic year, and subsequent class projects have been based on this model. The laboratory apparatus can also serve as a control systems design project for introductory and advanced control theory courses.

Overview. In this paper, we describe the SPENDULAP and illustrate its use in teaching and reinforcing key concepts in kinematics, dynamics, control, and simulation. The paper is organized as a set of independent sections, each of which illustrates an educational application of the SPENDULAP. This organization exploits the hyperlink capabilities of the HTML medium by providing easy navigation and cross referencing. In particular, note that at the bottom of the screen there is a small window that displays bibliographical references and cross-referenced display equations. In the remainder of this introduction, we briefly describe the SPENDULAP and then provide an overview of each of the topics covered in the other sections of the paper.

Brief description. The SPENDULAP, shown in Figure 1, consists of a free-swinging rigid pendulum that also rotates about an axis perpendicular to its swing axis. The pendulum is mounted in a motor-drive frame inside an acrylic housing. The pendulum features an initial position mechanism that allows one to initiate frame rotation with the pendulum at an arbitrary deflection about the swing axis. Transducers measure pendulum swing deflection and frame rotation rate. The drive motor can be controlled using a personal computer. A detailed description of the SPENDULAP can be found in System Description.

Kinematics Analysis. The kinematics of the SPENDULAP motion are fairly easy to visualize, but somewhat difficult to model due to coupling of two forms of rotational motion. The kinematics analysis allows students to gain proficiency and confidence using different rotating coordinate frames and cases of general motion. In Kinematics Analysis, we present a summary of a framework for kinematics analysis and then apply that framework to the SPENDULAP using two different rotating coordinate frames.

Dynamics analysis. The SPENDULAP is quite interesting from the viewpoint of dynamics analysis. The students can derive equations of motion using both Newtonian and Lagrangian approaches. Furthermore, the problem illustrates clearly the advantages and disadvantages of each approach. For example, the students are required to compute the internal forces on the swivel pin, and these forces are not readily available from the Lagrangian formulation. On the other hand, the students get an excellent appreciation of the ease in deriving equations of motion using the Lagrangian formulation as opposed to the Newtonian formulation. Newtonian Dynamics presents the Newtonian formulation of dynamics for the SPENDULAP, and Lagrangian Dynamics presents the Lagrangian formulation.

Nonlinear analysis. Many of the topics in nonlinear systems analysis initially appear to students to be abstract mathematical concepts with little physical meaning. The SPENDULAP is quite useful in allowing students to see the physical manifestations of those concepts. For example, if the motor is controlled such that the frame rotates at a constant angular velocity, the pendulum swinging motion can be simply treated as a second-order dynamical system. This system is nonlinear and is easily amenable to nonlinear analysis techniques such as linearization and state portraits. In Nonlinear Analysis, we illustrate the techniques of linearization and the state portrait using the SPENDULAP.

Simulation and animation. Simulation is an important part of an advanced dynamics course, and the SPENDULAP is well-suited to this topic as well. The equations of motion are straightforward and easily implemented in any programming language. Additionally, the SPENDULAP geometry is simple enough that students can produce graphical animations with a modest amount of effort. These animations help visualize the motion. Students find it particularly rewarding to compare their simulations with the motion of the laboratory apparatus. In Simulation we present a Simulink model of the SPENDULAP, and illustrate its use as an aid to visualizing the system kinematics and dynamics, and as an aid to extend the knowledge of the system dynamics gained from the Nonlinear Analysis.

Feedback control. A final use of the SPENDULAP is as a control system design problem. The control task is to design a feedback control law which adjusts drive motor torque to cause the pendulum to stabilize at a preselected deflection angle. This problem is nonlinear, but it can be solved using linear systems techniques. Of course, it is also quite suitable for nonlinear control techniques such as input-state linearization and sliding-mode (variable-structure) control. In Control we illustrate all three of these control approaches using the SPENDULAP example.