## Applied Control Theory Research

Caltech has an outstanding program in
control and dynamical systems . However, several problems in robotics
(in particular,
locomotion) involve significant problems in control theory. Hence,
even non-specialists in control must occasionally get involved in control
theory research. Currently, our efforts are focused in the following
areas:

### Control on Stratified Sets.

We are working on the extension of
standard notions of nonlinear controllability, trajectory generation, and
feedback to * stratified * systems. Systems, such as legged robots,
have a configuration space that has a naturally stratified structure.
Unfortunately, nonlinear control theory results do not hold for the systems
because of the discontinuous nature of their dynamics. We term this work
"applied" because the basic notions of nonlinear controllability have
already been established. However, this work is categorized as control
"theory" because the convential notions need a new theoretical framework
in order for them to be successfully extended to these new domains.

#### People working in this area:

### Motion Planning on Principal Bundles.

Many interesting mechanical systems, such as most locomotion systems,
have governing equations that evolve on a principle fiber bundle. We
are using tools from differential geometry and nonlinear control theory
to develop local motion planning techniques for this class of systems. The
key idea is to exploit the geometric structure inherent in this problem.
#### People working in this area:

### Hybrid Control Theory.

Broadly speaking, * hybrid * systems have properties that can be
characterized by both continuous (o.d.e. and p.d.e) and discrete (finite state
machine, finite automata) dynamical systems. For example, the stratified
systems described above are one example of a hybrid system. Due to the
widespread use of computers to control physical devices, hybrid systems
abound. The analysis of such systems lies at the boundary of computer science
and control theory. Successful hybrid system must robustly combine high-level
planning (planning of the finite automaton transitions) with feedback control
(evolution of the dynamic equations). While the planning and control problems
have been addressed separately by computer scientists and control theorists,
the interaction of the discrete and continuous worlds is largely left to the
ingenuity of engineers. As the complexity of the computer controlled physical
systems grows, such methods break down and there is a need for rigorous theory
to design, build, and evaluate hybrid systems.
With the support of a National Science Foundation post-doctoral fellowship
and a MURI grant, we
hope to contribute to the emerging hybrid control literature in the area of
control design for systems whose physical dynamics are hybrid and whose desired
behavior can be defined by a hybrid system. For more on hybrid systems,
see this page .

#### People working in this area:

### Switched Systems.

Switched systems are a special class of hybrid systems. Switching in
a control system can occur in the physics of the problem (i.e.,
a system changes contact state), or in the design of the controller. We
focus on problems where the switching occurs in the basic physical system
that is being modelled and controlled. For example, one can show that
overconstrained wheeled vehicles (such as the Sojourner vehicle of the
Mars Pathfinder mission) and distributed manipulation systems inherently
have switching mechanics. Our goal

#### People working in this area: