Collaborative Research: Dynamics, Geometry, and Control of Constrained Mechanical Systems

研究人员计划分析受约束的机械系统的动力学和控制。提出的研究包括：LR系统(具有左不变拉格朗日约束和右不变约束系统)的动力学和稳定性的研究，非完整积分器的性质，内部自由度控制系统的导航和镇定问题，变分积分器在受控拉格朗日方法中的应用，无限维非完整系统的动力学，可积非完整系统，以及超定坐标在力学和控制中的应用。研究人员还计划使用几何方法开发一种系统的方法来构造相空间中的坐标，该方法考虑到被研究系统的内在属性，从而使研究人员能够以简单而紧凑的方式写下运动方程。这种方法对于理解系统的动力学和进行考虑力学性质的数值模拟都是有帮助的。带约束系统的动力学在各种工业和科学应用中都是重要的。这类约束的例子包括滚动和滑动、链式刚性连杆、刚体上的转子动力学以及弹性杆和刚体之间的耦合。在工业、工程和科学中，出现这种限制的例子不胜枚举：机器人、轮式车辆的动力学和卫星在太空中的运动就是例子。在应用中，通常需要稳定稳定的稳态运动(如恒定速率的直线运动或圆周运动)--例如在实现所需的机器人或自主车辆运动时。我们的方法应该有助于实现这种运动，并总体上有助于分析和规定运动或机器人和自动车辆。我们还将提供一个框架，这将有助于开发此类系统的计算机模拟。

The investigators plan to analyze the dynamics and control of constrained mechanical systems. Proposed research includes the study of the dynamics and stability of LR systems (systems with left-invariant Lagrangian and right-invariant constraints), the properties of nonholonomic integrators, navigation and stabilization problems for systems with controls applied to internal degrees of freedom, applications of variational integrators to the method of controlled Lagrangians, dynamics of infinite-dimensional nonholonomic systems, integrable nonholonomic systems, and the use of overdetermined coordinates in mechanics and control. The investigators also plan to use geometric methods to develop a systematic approach to the construction of coordinates in the phase space that take into account intrinsic properties of a system under investigation and thus allows the researcher to write down equations of motion in a simple and compact way. This approach should be helpful both in understanding the dynamics of a system and in doing numerical simulations which respect mechanical properties.The dynamics of systems with constraints is important in various industrial and scientific applications. Examples of such constraints include rolling and sliding, chained rigid links, rotor dynamics on rigid bodies, and coupling between elastic rods and rigid bodies. There are numerous instances in industry, engineering and science where such constraints arise: robotics, the dynamics of wheeled vehicles, and the motion of satellites in space are examples. In applications, stabilization of steady-state motions (such as the straightforward or circular motion at a constant rate) is often desired -- for example in achieving a desired robotic or autonomous vehicle motion. Our methods should be helpful in achieving such motions and in general in analyzing and prescribing the motions or robotic and autonomous vehicles. We will also provide a framework which will be helpful in developing computer simulations of such systems.