Control of underactuated mechanical systems

该研究项目旨在系统地设计欠驱动非线性机械系统的控制设施，即执行器(控制输入)比机械自由度少的系统。该项目被细分为分析、设计和实施三个部分，将按照这种顺序进行工作，并具有预期的并行性。分析部分一方面从控制理论的角度对欠驱动系统进行分类，即根据输入对动力学的影响进行分类。因此，一个可能的工具是以合适的范式表示系统。另一方面，研究了固有惯性耦合的滚动以及离散对称和连续对称。此外，还分析了基于轨道平坦度的方法的适用性，即利用适当的时间变换获得微分平坦系统。欠驱动系统的一个中心特征是符合或违反所谓的Brockett条件，该条件表明存在渐近稳定平衡点的连续可微控制律的必要条件。项目第二部分的主题是针对这两种情况的控制算法的设计。作为对应用标准方法时出现的困难的反应，各种不同的方法被提出。在许多情况下，需要平衡点之间的过渡，这需要计算和稳定参考弹道。为了避免平衡点附近的不利可控性，有可能将运动设计为以零速度但非零加速度到达所需位置。如果这种运动是周期性的，就有可能考虑极限环的镇定，而不是一般的轨迹。对于违反Brockett条件的系统，控制器设计的进一步可能性包括状态空间的子流形的稳定化(而不是孤立平衡)和不连续控制律的形成。在项目的实施部分，所开发的分析和设计方法的范围是尽可能通用的，在现有的试验台上进行了实验验证。此外，为了简化所开发的方法和算法对控制界的适用性，计划为开放源代码的计算机代数系统提供一个软件工具箱。

The proposed research project aims at the systematic design of control facilities for underactuated nonlinear mechanical systems, i.e., systems with fewer actuators (control inputs) than mechanical degrees of freedom. The project is subdivided into the three parts analysis, design and implementation, which will be worked on in this ordering, with some expected parallelisms.The analysis part on the one hand deals with the classification of underactuated systems from the viewpoint of control theory, i.e., in terms of the influence that is exerted by the input to the dynamics. Thereby, a possible tool is the representation of the systems in a suited normal form. One the other hand the roll of the inherent inertial coupling as well as discrete and continuous symmetries is investigated. Furthermore, the applicability of approaches based on orbital flatness are assayed, i.e., obtaining a differentially flat system using a suited transformation of time.A central feature of underactuated systems is the compliance or violation of the so called Brockett condition, which states necessary conditions for the existence of a continuous differentiable control law asymptotically stabilizing an equilibrium point. Subject of the second part of the project is the design of control algorithms for both cases. As a reaction to the difficulties occurring when standard methods are applied, various different approaches are proposed. In many cases the transition between equilibrium points is desired, requiring the computation and stabilization of a reference trajectory. To avoid adverse controllability properties near equilibrium points it is possible to design the motion such that desired positions are reached with vanishing velocity but nonzero acceleration. If this motion is performed periodically, it is possible to consider the stabilization of a limit cycle instead of a general trajectory. Further possibilities for controller design for systems violating the Brockett condition include the stabilization of submanifolds of the state space (instead of an isolated equilibrium) and the formulation of discontinuous control laws.In the implementation part of the project the developed analysis and design methods, whose scope is aimed to be as general as possible, are experimentally verified on existing test beds.Additionally, to ease the applicability of the developed methods and algorithms for the control community it is planned to provide a software toolbox for an open source computer algebra system.