## Though hovercraft are rarely able to compete economically with traditional models of transportation in most environments, the amphibious versatility of hovercraft have given them a role in specialized applications including search and rescue, emergency medical services, ice breaking and recreational activities. Hovercraft have also proven to be one of the most promising vehicles for future Arctic offshore transportation. One of the most important ice characteristics affecting mid-winter offshore hovercraft operations is the distribution of ice roughness features, either as isolated ridges or as extensive rubble fields. In this environment hovercraft require accurate route marking and positioning systems combined with suitable obstacle avoidance capabilities. In our work we investigate nonlinear mechanical models for hovercraft dynamics, issues related to controllability and design of constructive control algorithms for steering autonomous hovercraft. The goal is to develop a high performance, low cost motion control system for a hovercraft.

## In recent years a geometric approach to studying mechanical systems has led to a deeper understanding of their behavior, controllability aspects and in the design of feedback controls. Key to this has been the development in reduction theory that exploits the invariance of the dynamics to a group of transformations (the symmetry group). The existence of a symmetry group permits the dropping of the dynamics to a lower dimensional (reduced) space. Lagrangian reduction involves dropping the Euler-Lagrange equations to the quotient of the velocity phase space given by the symmetry group while Hamiltonian reduction involves projecting the Poisson bracket to the reduced (quotient) space which also inherits a Poisson structure. The complete dynamics can then be reconstructed from those of the reduced system. In particular if the transformation group is a solvable Lie group then the solutions can be reconstructed using quadratures. The geometric phase associated with a path in the reduced system describes the motion of the complete (lifted) system. To model the dynamics of the hovercraft we identify the position and orientation of the hovercraft with an element of the Lie group SE(2). The invariance of the kinetic energy Lagrangian and the forcing term (thrust) under the SE(2) action of translations and rotations is used to obtain a set of Poisson-reduced equations on T*SE(2)/SE(2) which is isomorphic to se(2)*, the dual of the Lie algebra of SE(2). Depending on the control authority available, we distinguish two versions of the problem which we call the

"jet-puck"problem and the"hovercraft"problem respectively. In the jet-puck problem the control enters linearly, and we show that, using only a scalar control we have controllability over se(2)*. In the hovercraft problem the controls enter nonlinearly. We study the structure of the reduced dynamics and use modern methods of geometric control to understand issues related to controllability, constructive control algorithms and feedback stabilization of the complete (unreduced) system.

## We have shown controllability and the lack of small time local controllability of the Lie-Poison reduced equations of the hovercraft. We observe that every orbit the drift vectorfield of the reduced equations is periodic. This along with the fact that the Lie algebra rank condition is satisfied at all points on se(2)* is used to show controllability. The study of the dynamics of the hovercraft has led to a more general result on the controllability of the Lie-Poisson reduced dynamics for a class of mechanical systems that include the hovercraft, spacecraft and the underwater vehicle.

## A deeper understanding of hovercraft dynamics and constructive control algorithms for trajectory tracking can be used to enhance tracking capabilities in the face of uncertainty (e.g. wind gusts, vibration modes). Hybrid control architectures can be designed to generate ``behaviors'' based on sensor driven triggers, to allow for ``on the fly'' mode selection and gain scheduling.

## We plan to integrate into an off-the-shelf hovercraft an on-board computer, 19.2 KBaud communication gear, inertial aided GPS (global positioning system) or differential carrier phase GPS and surveillance instruments including a CCD camera with RF capabilities. Empirical data from flight tests will be used to build a more appropriate model (combining first principles and empirical data) of hovercraft dynamics. This will also serve as a testbed for hybrid architectures and motion control strategies developed during the course of our research.