Computational Geometric Mechanics and its Applications to Geometric Control Theory

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The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation. Computational geometric mechanics aims to leverage novel discrete differential geometric tools and discrete analogues of Lagrangian and Hamiltonian mechanics to systematically discretize the geometric approach to mechanics, while preserving geometric structure at a discrete level, thereby providing an efficient method of obtaining qualitatively accurate simulations over long times. The goal of this project is to continue the development of computational geometric mechanics and to apply it to geometric control theory, which will yield numerical implementations of control algorithms that exhibit good long-time behavior.This research will provide rational design principles for the construction of accurate and efficient real-time automatic control of modern engineering systems, such as robotic arms, spacecraft, and underwater vehicles. This is particularly important, due to the trend towards autonomous space and underwater vehicle missions with long deployment times and low energy propulsion systems, wherein accurate control algorithms are necessary to maximize the operational lifespan and range of these missions. Such missions will serve as the backbone of distributed space and underwater sensor networks that will enable us to continually monitor our oceans, environment, and climate.