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Current Research of Dr. J. William Goodwine

Stratified Control Systems

Many interesting and important control systems evolve on stratified configuration spaces. Roughly speaking, a configuration manifold is stratified if it contains submanifolds upon which the system is subjected to additional constraints or has different equations of state. For such systems, the equations of motion on each submanifold may change in a non-smooth, or even discontinuous manner, when the system moves from one submanifold to another. In such cases, traditional nonlinear control methodologies are inapplicable because they generally rely upon differentiation in one form or another. Yet it is the discontinuous nature of such systems that is often their most important characteristic because the system must cycle through different submanifolds to effectively be controlled. Therefore, it is necessary to incorporate explicitly into control methodologies the non-smooth or discontinuous nature of these systems.

Robotic systems, in particular, are of this nature. A legged robot has discontinuous equations of motion near points in the configuration space where each of its "feet'' come into contact with the ground, and it is precisely the ability of the robot to lift its feet off of the ground that enables it to move about. Similarly, a robotic hand grasping an object often cannot reorient the object without lifting its fingers off of the object. Despite the obvious utility of such systems, however, a comprehensive framework in which to consider control issues for such systems does not exist.

The fundamental approach of this work has been to exploit the physical geometric structure present in such problems to address control issues such as nonlinear controllability, trajectory generation and stabilization. The fundamental philosophy is to generate general results, i.e., results independent of a particular robot's number of legs, fingers or morphology.

Unstable Rolling Dynamics and Shimmy

Mechanical systems which contain rolling elements are naturally modeled as nonholonomic mechanical systems. However, the "rolling without slipping" assumption is clearly an approximation and is not valid, for example, for elastic rolling contact above a certain speed. Additionally, for real physical rolling systems, the rolling without slipping constraint is imposed by friction. However, since a friction force has a limited magnitude, if the nonholonomic constraint force exceeds that limit, the real rolling system will transition from a rolling state to a skidding state. Unstable rolling is obviously a very important phenomenon in vehicle dynamics, affecting many systems from aircraft nose wheels, truck trailers and motorcycle front wheels to the ubiquitous shimmying shopping cart wheel. This research is directed toward the design and analysis of "switching" type controllers for such systems.

Vision-Based Robotic Control

This research is directed toward extending results in Camera Space Manipulation ("CSM") to nonholonomic systems. The application of vision-based control strategies to robotic systems has been hindered by problems inherent in image analysis and calibration. CSM is a demonstrated effective, open loop, estimation based control strategy that avoids these difficulties in image analysis and calibration. However, to date, the primary application of CSM has been to holonomic systems. Nonholonomic systems are generally more difficult than holonomic systems, and these difficulties carry over to the CSM context as well. Another difficulty with extending CSM to the nonholonomic case is that if the cameras are mounted on a nonholonomic robotic base the target location in the camera image planes is no longer stationary.