Collaborative Research: Control of Underactuated Mechanical Systems through Lyapunov Direct Methods

合作研究：用李雅普诺夫直接方法控制欠驱动机械系统该项目涉及发展李亚普诺夫直接方法来设计非线性欠驱动机械系统的镇定控制器，使系统成为闭环李雅普诺夫形式而不是闭环哈密顿形式。目标是证明与闭环李亚普诺夫形式相关的控制器可以通过构成渐近稳定性的某些线性(偏)微分方程组的解析解或数值解(在反馈中完成)来实现。所提出的方法的优点是哈密顿形式的系统类是Lyapunov形式的系统类的严格子集，从而产生更大类的镇定控制器，从而可以提供系统性能的改善。此外，预计拟议的研究将为使用其他方法处理难以解决的问题提供基础，同时还将导致利用受控拉格朗日和受控哈密顿方法的统一框架。另一个重要的结果是所得到的控制器关于速度是线性的，而现有的基于能量的非线性控制器关于速度是二次的。将用该方法设计的控制律与用受控拉格朗日方法、拉格朗日方法和IDA-PBC方法设计的控制律进行比较，主要目的是分析和进一步发展一种新的策略--Lyapunov直接方法，用于设计欠驱动机械系统的稳定控制器，该系统的特征是驱动比自由度少的系统，例如一个或多个关节没有动力的机器人。这类系统的例子有航天器、水下航行器、卫星、气垫船和集装箱船的货运起重机。该方法对现有技术进行了改进，在不存在现有方法的一些常见问题的情况下，获得了更好的性能。预计有一些系统可以用提议的技术治疗，而目前的技术不能治疗。分析将包括李亚普诺夫直接法和现有方法产生的控制律的比较。将通过就业和课程学分鼓励数学和工程专业的本科生参与该项目。一些学生将帮助KSU的REU项目的发展。为了促进代表不足的群体的参与，科罗拉多州立大学少数民族工程计划和工程和科学计划中的女性办公室承诺帮助招生，这一活动与他们由NSF资助的STEP计划相吻合。

AbstractCollaborative Research: Control of Underactuated Mechanical Systems through Lyapunov Direct MethodsThe proposed project involves developing Lyapunov direct methods for designing stabilizing controllers for nonlinear underactuated mechanical systems that render the system into closed loop Lyapunov form as opposed to closed loop Hamiltonian form. A goal is to show that the controller associated with the closed loop Lyapunov form can be implemented through either an analytical or a numerical solution (done in the feedback) of certain linear (partial) differential equations constituting asymptotic stability. The advantage of the proposed methodology is that the class of systems in Hamiltonian form is a strict subset of the class of systems in Lyapunov form thus yielding a larger class of stabilizing controllers that could offer improvements in system performance. It is further expected that the proposed research will provide a basis on which to treat problems that are intractable using other methods, while also leading to a unifying framework for methods utilizing controlled Lagrangians and controlled Hamiltonians. Another significant result is that the obtained controller is linear with respect to the velocities, whereas existing energy based nonlinear controllers are quadratic in the velocities. Comparisons between control laws designed through the proposed method and those designed through controlled Lagrangian, lambda, and IDA-PBC methods will be made.The primary thrust of the proposed activity is to analyze and further develop a new strategy, Lyapunov direct methods, for designing stabilizing controllers for underactuated mechanical systems, systems characterized by having fewer drives than degrees of freedom such as a robot where one or more joints are not powered. Examples of such systems are spacecraft, underwater vehicles, satellites, hovercraft and cargo cranes for container ships. The proposed method offers an improvement over the current art in that better performance is achieved while not suffering some common problems of existing methods. It is expected that there are systems that can be treated by the proposed techniques that cannot be treated by current techniques. The analysis will include comparisons of control laws produced by Lyapunov direct methods and current published methods. Undergraduate students in both mathematics and engineering will be encouraged to participate in the project through employment and course credit. Some students will help with development of an REU program at KSU. To facilitate participation of underrepresented groups, the KSU Minority Engineering Program and Women in Engineering and Science Program offices have committed to help in student recruitment, an activity that dovetails with their NSF funded STEP program.