Computational Geometric Mechanics and its Applications to Geometric Control Theory

在几何控制理论中，力学的几何方法是创新控制方法的理论基础。这些技术允许通过改变卫星的形状来控制卫星的姿态，而不是化学推进，这是理解一只下落的猫总是用脚着地的能力的基础，即使在反向释放时也是如此。计算几何力学旨在利用新颖的离散微分几何工具以及拉格朗日和哈密顿力学的离散类似物，将力学的几何方法系统地离散化，同时在离散水平上保留几何结构，从而提供一种长时间获得定性精确模拟的有效方法。该项目的目标是继续发展计算几何力学，并将其应用于几何控制理论，这将产生具有良好长期性能的控制算法的数值实现。本研究将为机械臂、航天器、水下航行器等现代工程系统构建精确、高效的实时自动控制提供合理的设计原则。这一点尤其重要，因为自主空间和水下航行器任务具有长部署时间和低能量推进系统的趋势，其中需要精确的控制算法来最大限度地延长这些任务的使用寿命和范围。这些任务将成为分布式空间和水下传感器网络的骨干，使我们能够持续监测我们的海洋、环境和气候。

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.