Modeling and Control of Closed Kinematic Chains

文摘：闭式运动链机械系统与开式链式机械系统相比，具有运动惯量小、运动速度快、刚性大、力重比高等优点。然而，由于运动学、动力学和控制分析的复杂性，适用于控制器设计的闭链机构的动力学方程的推导仍然是一个挑战。这项研究确定了这些系统的建模和控制中的主要困难，并提供了一个分析框架来推导适用于高级基于模型的控制律的动力学模型。研究了两种面向控制的建模方法。在第一种方法中，用独立的广义坐标(简化模型)建立了闭链机构的动力学方程。考虑简化模型的局部性和隐含性质，将开链的丰富控制律推广到闭链。第二种方法是将描述闭合运动链的微分代数方程(DAE)转化为具有渐近稳定快动力学的奇异摄动微分方程组。结果表明，当对奇异摄动系统进行适当的控制设计时，当人工引入的小参数较小时，受控奇异摄动系统的响应将与受控DAE系统(闭链机构)的响应非常接近。本研究中开发的技术可能适用于DAE描述的许多其他系统。本研究的教育目的是帮助在传统的动力学和控制课程和实验室中介绍复杂的闭链动力系统的动态建模和控制。特别是，将在万维网上使用机器人装置进行控制实验，以便更有效地传播这项研究的结果，并将其纳入控制教育。

Abstract: Closed kinematic chain mechanical systems promise advantages over open chains in terms of less moving inertia, faster and stiffer motion, and much better force-to-weight ratio. However, derivation of dynamic equations of motion for closed chain mechanisms suitable for controller design is still a challenge because of the complexity of the kinematics, dynamics, and control analyses. This research identifies the central difficulties in the modeling and control of these systems and provides an analytical framework to derive dynamics models appropriate for advanced model-based control laws. Two modeling-for-control approaches are investigated. In the first approach, a formulation of the dynamics equations of closed chain mechanisms in terms of independent generalized coordinates (reduced model) is developed. Methods to extend the wealth of control laws of open chains to closed chains while taking into consideration the local and implicit nature of the reduced model are considered. The second approach is to transform the Differential Algebraic Equations (DAE) describing closed kinematic chains into a singularly perturbed differential equation system with asymptotically stable fast dynamics. It follows that when a proper control design is made for the singularly perturbed system, then for a small value of an artificially introduced small parameter, the response of the controlled singularly perturbed system would be quite close to that of the controlled DAE system (closed chain mechanism). The techniques developed in this research potentially have applications to many other systems described by DAE. The educational objective of this research is to help introduce dynamic modeling and control of complex closed chain dynamical systems in the traditional dynamics and control courses and laboratories. In particular, a robotic device will be made accessible for control experimentation over the World Wide Web for a more effective dissemination of the results of this research and integration in control education.