Control and Stabilization of Mechanical Systems with Broken Symmetry via Symmetry Recovery

许多机械或机器人系统被设计成在特定配置或位置中操作。例如，水下航行器必须保持一定的姿态以执行其任务，例如水下监视或海上风力涡轮机必须在海洋中的波浪和阵风产生的各种干扰下保持其直立位置。然而，这种配置在外部干扰下往往是自然不稳定的，因此必须适当地控制以保持在期望的配置中。该项目的主要目标是创新新的控制策略，以稳定和控制这些难以控制的系统，利用这些系统的一些固有的动态特性。新的控制策略将适用于广泛的一类机械和机器人系统，而不是目前使用的专门为特定系统设计的技术。该项目涉及一群来自不同背景和学术水平的学生，并使用简单易懂的机器人系统将研究与外联活动结合起来。 在重力或浮力影响下的机械和机器人系统可以被视为对称性破缺的系统：外力的存在引入了一个特定的方向，力沿着该方向施加到系统上，从而打破了系统原本拥有的对称性。通过控制来平衡或稳定这样的系统一直是控制理论的主要挑战之一。该项目开发了一个新的范例的控制和稳定的机械系统的对称性破缺。该项目以对称性破缺的拉格朗日/哈密尔顿系统的微分几何公式为基础，旨在开发适用于重力/浮力影响下的机械和机器人系统的一般稳定理论。该项目通过系统地将受控拉格朗日/哈密顿方法扩展到对称性破缺的系统，扩大了为机械系统开发的稳定技术的应用范围。该方法也将被推广到非完整约束下的机械系统，进一步扩展应用范围到具有滚动和滑动约束的机器人系统以及具有速度约束的水下航行器。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many mechanical or robotic systems are designed to operate in a specific configuration or position. For example, an underwater vehicle must maintain a certain attitude to carry out its missions such as underwater surveillance or an offshore wind turbine must maintain its upright position under various disturbances created by waves and gusts in the ocean. However, such a configuration tends to be naturally unstable under external disturbances, and hence must be controlled properly to stay in a desired configuration. The main objective of this project is to innovate new control strategies to stabilize and control such difficult-to-control systems, exploiting some inherent dynamic characteristics of these systems. The new control strategies will be applicable to a wide class of mechanical and robotic systems as opposed to ad-hoc techniques that are currently used which are designed for a specific system. The project involves a group of students from diverse backgrounds and academic levels, and integrates research with outreach activities using a simple and easy to understand robotic system. Mechanical and robotic systems under the influence of gravity or buoyancy can be seen as systems with broken symmetry: The presence of the external forces introduces a particular direction along which the force applies to the system, thereby breaking the symmetry that the system would otherwise possess. Balancing or stabilizing such systems by applying controls to them has been one of the main challenges in control theory. The project develops a new paradigm of control and stabilization of mechanical systems with broken symmetry. Building on the differential-geometric formulation of Lagrangian/Hamiltonian systems with broken symmetry, the project aims to develop a general theory of stabilization that applies to mechanical and robotic systems under the influence of gravity/buoyancy. The project broadens the scope of applications of stabilization techniques developed for mechanical systems by systematically extending the method of controlled Lagrangians/Hamiltonians to systems with broken symmetry. The methods will also be generalized to those mechanical systems under nonholonomic constraints as well, further extending the scope of applications to those robotic systems with rolling and sliding constraints as well as underwater vehicles with velocity constraints.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.