Workshop on Geometric Control of Mechanical Systems

The geometric theory of control for mechanical systems has reached a quite mature state, as is evidenced by the recent appearance several monographs on the topic. The workshop is an outgrowth of a book written by the organizers. The primary emphasis of this workshop is the modeling, analysis, and control of mechanical systems. The methods and results presented can be applied to a large class of mechanical control systems, including applications in robotics, autonomous vehicle control, and multi-body systems. The workshop presents a unified treatment of parts of control theory for mechanical systems. A distinctive feature of the presentation is its reliance on techniques from differential and Riemannian geometry. This workshop will outline the areas of overlap between geometric mechanics and control theory for mechanical systems. Liberal use of examples form an integral part of the presentation. The workshop begins with mathematical background, motivated through innovative approaches to physical modeling, analysis, and design techniques.

The following describes the target audience for the workshop.

1. Graduate students: Students in engineering (e.g., robotics, control, dynamical systems) will get an idea of the sort of background they will need to apply the tools we discuss. Students in mathematical sciences will get exposure to interesting problems in control that rely on some fairly sophisticated mathematics. Both groups of students will learn about some open problems in the area.
2. Researchers in application areas: Researchers will be exposed to systematic, implementable techniques for solving difficult problems in control theory.
3. Researchers in mechanical systems theory: Part of the discussion will be directed towards open problems in the field. In this way, the workshop will be useful, even for specialists in the area, and hopefully some fruitful discussions will evolve on future directions in the field.
4. Teachers: Some teaching material will be made available at the workshop to give teachers some ideas of ways the material can be effectively presented to students.

[Confirmed speakers] [Workshop schedule] [Course material]

Confirmed speakers

Francesco Bullo received the Laurea degree ``summa cum laude'' in Electrical Engineering from the University of Padova, Italy, in 1994, and the Ph.D. degree in Control and Dynamical Systems from the California Institute of Technology in 1999. From 1998-2004 he was an Assistant Professor with the Coordinated Science Laboratory at the University of Illinois at Urbana-Champaign. He is currently an Associate Professor with the Mechanical & Environmental Engineering Department at the University of California, Santa Barbara. His research interests include motion planning and coordination for autonomous vehicles, and geometric control of mechanical systems.

Jorge Cortés received the Licenciatura degree in mathematics from the Universidad de Zaragoza, Spain, in 1997, and the Ph.D. degree in engineering mathematics from the Universidad Carlos III de Madrid, Spain, in 2001. From January to June 2002, he held a postdoctoral position at the Systems, Signals and Control Department of the University of Twente. He is currently a Postdoctoral Research Associate at the Coordinated Science Laboratory of the University of Illinois at Urbana-Champaign. His current research interests focus on mathematical control theory and geometric integration, with a special emphasis on Lagrangian and Hamiltonian systems and the role of symmetry principles, and motion coordination algorithms for groups of autonomous vehicles.

Andrew Lewis received his undergraduate degree in Mechanical Engineering from the University of New Brunswick in 1987, and his MSc and PhD in 1988 and 1995, respectively, both in Applied Mechanics from the California Institute of Technology. From 1995-1996 he was a Postdoctoral Fellow in Control and Dynamical Systems at the California Institute of Technology, and from 1996-1998 he was a Postdoctoral Fellow in the Mathematics Department at the University of Warwick. He is now an Associate Professor in the Mathematics and Statistics Department at Queen's University in Kingston, Ontario in Canada. His research interests include geometric mechanics and differential geometric control theory.

Sonia Martínez received the Licenciatura degree in mathematics from the Universidad de Zaragoza, Zaragoza, Spain, in June, and the Ph.D. degree in engineering mathematics from the Universidad Carlos III de Madrid, Madrid, Spain, in 2002. From September 2002 to September 2003, she held a postdoctoral position at Department of Applied Mathematics IV of the Universidad Politecnica de Catalunya. From October 2003, she is a Postdoctoral Fulbright Fellow at the Coordinated Science Laboratory of the University of Illinois at Urbana-Champaign. Her current research interests include optimal control policies for robotic locomotion, controllability analysis and motion planning for underactuated systems, and low-complexity representations of mechanical systems.

Workshop schedule

Time	Topic	Speaker
Morning (½ hour)	Problem descriptions and motivation	Andrew D. Lewis
Morning (3 hours)	Geometric modeling	Andrew D. Lewis
Afternoon (1 hour)	Controllability	Francesco Bullo
Afternoon (1 hour)	Kinematic reduction and motion planning	Francesco Bullo
Afternoon (1 hour)	Perturbation methods and oscillatory stabilization	Sonia Martínez and Jorge Cortés
Afternoon (½ hour)	Open questions	Organizers

Details on topics

1. Problem descriptions and motivation: Examples will be used to present an overview of what will be covered during the course of the workshop. Problems discussed will include those of modeling, controllability, motion planning, and stabilization.
2. Geometric modeling: One of the major hurdles in applying the techniques presented in the workshop is the differential geometric formalism for mechanics. This portion of the workshop will be devoted to a down to earth discussion of the mathematics involved in the geometric modeling of mechanical systems. The objective will be to understand, in as concrete a way as possible, how one goes from a physical problem description to the components of the geometric models used in the control analysis and design.
3. Controllability: One of the interesting features about mechanical control systems is the interplay of the geometry of the system models and the geometry of nonlinear control theory. In this portion of the talk, this interplay will be examined in the context of controllability.
4. Kinematic reduction and motion planning: This part of the workshop continues the controllability analysis, but now concentrating attention on systems whose controllability is determinable to low-order. Such systems exhibit some remarkable connections between controllability and motion planning that will be illustrated on some nontrivial examples.
5. Perturbation methods and oscillatory stabilization: This part of the workshop focuses on the analysis of mechanical systems subject to oscillations and on the design of control schemes that have large amplitude and large frequency.
6. Open questions: Possible open problems include (1) the geometric analysis of singularities in modeling of mechanical systems with constraints, (2) the design of provably correct ``locomotion gaits,'' that is, of motion planning algorithms based on the weakest possible controllability conditions. Other topics will arise as per the interests of the participants.

1. Slides used during the workshop.
2. Material on CD distributed during workshop [Gzipped tar file (.tgz)] [Zip file (.zip)].
3. The workshop is based on the material in the organizers' book Geometric Control of Mechanical Systems. This book will be in print at the time of the workshop, and will be available for purchase.
4. In class lecture notes used at Queen's University.
5. Slides from CMU talk by Francesco.