We study the use of a nonlinear projection operator in the development of a novel function space approach to the optimization of trajectory functionals. Given a bounded state-control trajectory of a nonlinear system, one may make use of a simple (e.g., linear time-varying) trajectory tracking control law to explore the set of nearby bounded state-control trajectories. Such a trajectory tracking control system defines a nonlinear projection operator that maps a set of bounded curves onto a set of nearby bounded trajectories.
The projection operator approach provides a means for developing a Newton descent method for the optimization of dynamically constrained functionals. By projecting a neighboring set of state-control curves onto the trajectory manifold and then evaluating the cost functional, the constraint imposed by the nonlinear system dynamics is subsumed into an unconstrained trajectory functional. This equivalent optimization problem may now be attacked in an essentially unconstrained manner leading to an algorithm defined in function space that produces a descending sequence in the Banach manifold of bounded trajectories.
This approach can be extended to optimal control problems with inequality constraints (state, control, and mixed) through the use of (extended) barrier functionals of log type. Intuitively, one can see that the resulting interior point methods might be effective for convex inequality constraints. It is thus somewhat surprising that these methods are also quite effective for the decidedly nonconvex constraints that arise in collision-free maneuver planning for cooperating vehicles. Computational results for a number of constrained trajectory optimization problems, including minimum time and minimum energy, will be discussed.
John Hauser received the BS degree from the United States Air Force Academy in 1980 and the MS and
PhD degrees from the University of California at Berkeley in 1986 and 1989, all in Electrical Engineering
and Computer Science. From 1980 to 1984, he flew Air Force jets throughout the United States and
Canada participating in active Air Defense exercises. In 1989, he joined the Department of EE-Systems
at the University of Southern California as the Fred O’Green Assistant Professor of Engineering. Since
1992, he has been at the University of Colorado at Boulder in the Department of Electrical, Computer,
and Energy Engineering. He has held visiting positions at University of Padova, Caltech, Instituto
Superior Tecnico in Lisbon, Lund Institute of Technology, and Ecole Superieure d’Electricite. He received
the Presidential Young Investigator award from the National Science Foundation in 1991.
John Hauser’s research interests include nonlinear dynamics and control, optimization and optimal
control, aggressive maneuvering for high performance motorcycles and aircraft and other vehicles, and
dynamic visualization. Recent work has focused on the development of optimization (and optimal
control) tools and techniques for trajectory exploration with an eye toward characterizing the trajectory
space (with limitations) of highly maneuverable nonlinear systems. This work finds application in the
control of highly configurable UAVs (with propulsion vectoring) and in the analysis of racing motorcycles.
