Dynamics, Integrability and Control of Mechanical and Nonholonomic Systems

机械和非完整系统的动力学、可积性和控制

• 批准号: 1207693
• 负责人: Anthony Bloch
• 金额: $ 17万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207693&HistoricalAwards=false
• 关键词: Dynamics; Integrability; Control; Mechanical; Nonholonomic

This project is a continuation of the principal investigator's study of the geometry, dynamics and control of mechanical systems including Hamiltonian and Lagrangian systems, nonholonomic systems, and gradient flows. Nonholonomic systems are a generalization of classical Hamiltonian and Lagrangian systems where the system is subject to nonintegrable constraints on the velocities. The investigator proposes to study the dynamics and control of mechanical systems with such constraints, including nonholonomic systems with internal degrees of freedom, nonholonomic systems with controls, systems with nonlinear constraints, the Hamilton Jacobi equation in the nonholonomic setting, discrete systems, infinite-dimensional systems, and certain optimal control problems. He also proposes to study integrable Hamiltonian and nonholonomic systems and their relation to gradient flows. In the course of this work the proposer will study the role that symmetries play in the formulation of the nonholonomic equations of motion and in their integrability, the new dynamics that arises when one has a nonholonomic system interacting with a fluid, and how systems with nonlinear nonholonomic constraints behave, including thermostats (systems interacting with a heat bath). He will also study flexible nonholonomic systems. and the role of Hamiltonization (transformation to Hamiltonian form) in understanding the integrability of a nonholonomic system. In addition he will analyze gradients flows in finite and infinite dimensions including flows on loops groups and other infinite-dimensional spaces, and their relationship to integrability.The dynamics of mechanical systems is of great importance in technology. The theory of nonholonomic dynamics in particular is the study of mechanical systems subject to constraints imposed on velocities. Such constraints arise for example in systems consisting of rigid bodies rolling on surfaces without slipping Nonholonomic systems occur frequently in many practical systems including wheeled vehicles such as cars (in particular self steering cars) and robots. Control of such systems is an important technological problem. In addition, the mathematics behind the control of nonholonomic systems plays a key role in control of nonlinear systems in general, such as the control of aircraft or underwater vehicles. Also important is how dissipation (or friction) affects the behavior and stability of such systems We are interested in describing the behavior of these systems, how to control and stabilize them and how to simulate the dynamics and control on a computer. It is hoped that this research will lead to advances in engineering and the proposer will collaborate with engineers and physicists. The proposed program has a strong educational impact. Material related to this research will be used in an advanced dynamics class. The research will involve the work of Ph.D students and undergraduates.

该项目是主要研究者对力学系统的几何、动力学和控制的研究的延续，包括哈密顿和拉格朗日系统、非完整系统和梯度流。 非完整系统是经典哈密顿系统和拉格朗日系统的推广，其中系统受到速度的不可积约束。 研究人员建议研究机械的动力学和控制 具有此类约束的系统，包括具有内部自由度的非完整系统、具有控制的非完整系统、具有非线性约束的系统、非完整环境中的汉密尔顿雅可比方程、离散系统、无限维系统和某些最优控制问题。他还建议研究可积哈密顿和非完整系统及其与梯度流的关系。 在这项工作的过程中，提议者将研究对称性在非完整运动方程的制定及其可积性中所起的作用，当一个非完整系统与流体相互作用时出现的新动力学，以及具有非线性非完整约束的系统如何表现，包括恒温器（与热浴相互作用的系统）。他还将研究柔性非完整系统。 以及哈密尔顿化（变换为哈密尔顿形式）在理解非完整系统可积性中的作用。此外，他将分析梯度流在有限和无限的维度，包括流动的循环组和其他无限维空间，以及它们的关系，可积性。力学系统的动力学是非常重要的技术。非完整动力学理论是研究受速度约束的力学系统。这种约束出现在例如由在表面上滚动而没有滑动的刚体组成的系统中。非完整系统经常出现在许多实际系统中，包括轮式车辆，例如汽车（特别是自转向汽车）和机器人。 控制这样的系统是一个重要的技术问题。 此外，非完整系统控制背后的数学在一般非线性系统的控制中起着关键作用，例如飞机或水下航行器的控制。 同样重要的是耗散（或摩擦）如何影响这些系统的行为和稳定性我们感兴趣的是描述这些系统的行为，如何控制和稳定它们，以及如何在计算机上模拟动力学和控制。 希望这项研究将导致工程的进步，提议者将与工程师和物理学家合作。 该计划具有很强的教育影响力。与本研究相关的材料将用于高级动力学课程。 研究将涉及博士生和本科生的工作。