无穷维非完整力学系统的对称约化、控制及其应用

Infinite-dimensional nonholonomic mechanical systems arising in many engineering applications in the context of mobile flexible and soft robotics are ubiquitous in the nature. The objectives of this programme are to construct high-precision numerical simulators and control algorithms for infinite-dimensional nonholonomic mechanical systems capable of performing calculations over exponentially-long time intervals and to give the related geometric framework of reduction and control in geometric mechanics. To this end, firstly, the generalization of classical.Hamel’s formalism for Lagrange and Hamilton field theories will be developed to give a flexible representation of the equations of motion of constrained field theories in which the Lagrange multipliers are intrinsically absent. Secondly, by using the proposed Hamel’s formalism, Stokes-Dirac structures in infinite-dimensional dynamical systems, multi-Dirac structures in field theories, and corresponding reduction theories will be explored. Thirdly, to reduce the complexity of numerical computations in optimal control, reduction of controlled systems in infinite-dimensional nonholonomic mechanical systems with symmetry will be investigated in two ways: through the reduction of Dirac structures describing Pontryagin's maximum principle and reduction of the Hamilton-Pontryagin variational principle for optimal control problems. An infinite-dimensional controlled Lagrangian method for stabilization of infinite-dimensional nonholonomic systems will be also studied. Finally, the structure-preserving algorithms for simulation and control of infinite-dimensional nonholonomic systems will be constructed. Certain particular examples, including formation control of unmanned ground vehicles, will be given to illustrate the effectiveness of the developed algorithms.

无穷维非完整力学系统源于移动柔性和软体机器人等工程应用，并普遍存在于自然界中。本项目旨在构建适用于无穷维非完整力学系统的能长时间保持高精度的数值模拟与控制算法和在几何力学中建立相关的约化与控制几何框架。为此，首先，将经典力学的Hamel形式推广到Lagrange场论和Hamilton场论， 从而给出约束场论内禀且无需Lagrange乘子的运动方程的灵活表达。其次，利用Hamel形式探讨无穷维动力系统的Stokes-Dirac结构和场论的多Dirac结构。再次，研究无穷维非完整力学系统的最优控制与镇定。为降低最优控制计算问题的复杂度，以两种方式研究具对称性非完整力学系统最优控制的约化：通过约化描述Pontryagin极大值原理的Dirac结构，及约化最优控制的Hamilton-Pontryagin变分原理。最后，研究适用于无穷维非完整系统的保结构算法，并给出无人车编队等实例阐明算法的有效性。

针对集群、移动软体机器人等工程应用需求，和对无穷维约束力学系统建模理论的需要，本项目针对无穷维非完整系统，首先建立了经典场论的 Hamel 形式理论， 其作为刻画约束力学系统运动的标准型能广泛地应用于对该系统的动力学建模与分析；利用经典场论Hamel形式实现了对无穷维非完整力学系统的对称性约化，为发展对系统长时间定性准确模拟和鲁棒性强的控制算法奠定了基础。通过提供了一个新的无穷维非完整力学系统的代表性例子——柔性雪橇（Chaplygin-Timoshenko雪橇）验证了所得理论的有效性。 . 其次，构建适用于无穷维非完整系统的数值模拟和控制的保结构算法——Hamel场变分积分子（空间一维情形），应用于典型无穷维非完整系统——Chaplygin-Timoshenko雪橇的仿真；以几何精确梁为例，探究了Hamel 场变分积分子的保动量性质；基于Hamel形式给出一种无条件稳定动力学积分算法；建立了对（非完整）约束力学系统能系统地构造高精度保结构算法的平台——非完整精确积分子；用经典场论Hamel形式，以几何精确Euler-Bernoulli梁为例探究了无穷维力学系统的约束实现，以期获得计算其动力学的快速算法。. 最后，基于Hamel形式得到多体系统和非完整系统最优控制与镇定（轨迹追踪）高效算法，并分别应用于工业和移动机器人；在场论Hamel形式框架下，提出柔性集群编队控制算法，可用于围捕等队形变换场景，在无人艇群编队控制的工程实践中得到成功的应用。