# Main Page

 Bill Goodwine Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN 46556 bill@controls.ame.nd.edu

# Research

## Papers

### Cyber Physical Systems

My research focuses primarily on theoretical nonlinear control with recent emphasis on Cyber Physical Systems. Cyber physical systems are systems with highly integrated physical and computational components (often involving the complication of networked communication). They tend to be very large and complex in scale. While many CPS systems exist in the real world, to date there are few general theoretical results available to guide both the design of such systems and the control of such systems. Most existing CPS systems are designed and controlled based upon accumulated real-world industrial knowledge that tends to be industry- or application-specific. My recent work has focused particularly on so-called symmetric systems. A symmetric system is comprised of many components with the restriction that the components be very closely related and connected together in a "regular" manner. With such restrictions, it is possible to formulate general models and then consider what types of properties remain invariant as components are added to or removed from the system. A related question is how the system behaves as components fail, which is a question of robustness.

### Stratified Systems

Many interesting and important control systems evolve on stratified configuration spaces. Roughly speaking, we will call a configuration manifold 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.

### Control of Mechanical Systems

Most theoretical control results are based upon very generic dynamical systems formulations, such as ${\displaystyle {\dot {x}}=Ax+Bu}$ for linear systems or ${\displaystyle {\dot {x}}=f(x)+g(x)u}$ for a nonlinear system. Of course this leads to the question of whether a more restrictive starting point can lead to valuable results. An important area of research along these lines is so-called control of mechanical systems where the equations of motion are not as general, but are assumed from the beginning to come from some first principle of mechanics. We have focused specifically on control of Lagrangian systems that are underactuated. Specifically, it is possible in such a framework to write general expressions for the relationship of the coupling between the controlled degree of freedoms and uncontrolled degrees of freedom, and given such expressions it is possible to know when there is close coupling between them and total decoupling between them. Furthermore, it is often the case that the coupling between the controlled and uncontrolled degrees of freedom is such that it may be only of one sign, i.e., no matter what is done with the control inputs, the uncontrolled degrees of freedom may only increase (or decrease) in magnitude. Such results have obvious important implications for control algorithms.

### Other Projects

Other smaller projects include:

• control of aero-optic systems
• predictive biosimulation for human metabolism
• fuzzy logic-based robust control for stratified systems
• model-predictive control for marine navigation.

# Biographical Sketch

• MS and PhD degrees in Applied Mechanics from the California Institute of Technology in 1993 and 1998, respectively.
• JD degree from Harvard Law School, 1991, cum laude
• Instructor, Assistant Professor, Associate Professor, Department of Aerospace and Mechanical Engineering, University of Notre Dame, 1998 - present.
• Associate Department Chair, Department of Aerospace and Mechanical Engineering, University of Notre Dame, August 2008 - August 2012.
• Member of the Illinois Bar Association, 1991 - present.
• Registered Patent Attorney, 1998 - 2004 (not maintained).
• NSF CAREER Award Recipient.
• Boeing Welliver Faculty Fellow.
• Dockweiler Award for Excellence in Undergraduate Advising, May 2010.
• BP Foundation Outstanding Teacher of the Year, College of Engineering, Spring, 2008.
• Joyce Award (teaching), Spring, 2008.
• University of Notre Dame Kaneb teaching award, Spring, 2005.
• Department of Aerospace and Mechanical Engineering Ruth and Joel Spira Award for Excellence in Teaching, 2003 - 2004 and 2007 - 2008.
• American Society of Engineering Education Illinois/Indiana Section Outstanding Teaching Award, April, 2003.
• Department of Aerospace and Mechanical Engineering Faculty Award (teaching), 1998 - 1999.

# Courses

### Course Blog

In order to be able to interactively answer questions online, I've maintained a course blog for all courses since 2002.