Collaborative Research: Ergodic Trajectories in Discrete Mechanics

这一研究项目的目标是开发高效和稳健的数值方法，使用机械系统搜索和探索环境，同时确保平均时间覆盖范围重现规定的空间分布。这是通过发展控制机械系统的方法来实现的，以使它们相对于给定的空间分布是遍历的，并将它们与具有良好的向后误差特性的保持几何结构的数值积分器相结合，并且保持几何不变量，如辛结构、能量和动量。此外，这些方法保留了配置流形的非线性结构，如李群或齐次空间结构。主要的技术目标包括：(I)开发和分析结构化积分器以准确预测给定系统的遍历特性；(Ii)开发基于模拟的系统参数优化和控制以最大化遍历搜索的效率；(Iii)将这些技术推广到李群和齐次空间，使得能够对丰富的机器人系统进行遍历搜索；(Iv)在现实系统上进行实验验证。这些技术将直接适用于广泛的现实世界工业应用，包括机器人系统的联合空间探索，制造中的故障检测，以及自主传感器网络的最佳搜索、覆盖和信息提取。将通过发布公共领域软件来促进这一工业扩展，这些软件将降低在各种应用中采用几何数字积分技术的障碍。此外，我们将通过与芝加哥科学和工业博物馆合作，开展公共推广活动，开发一个互动展览，展示机械系统的搜索算法。这些具体的申请有助于激励高中生(特别是代表不足的少数族裔和女性)攻读STEM学位，这反过来将有助于确保美国工业的长期经济创新和竞争力。

The objective of this research project is to develop efficient and robust numerical methods for searching and exploring an environment using a mechanical system, while ensuring that the average temporal coverage reproduces a prescribed spatial distribution. This is achieved by developing methods for controlling mechanical systems so that they are ergodic with respect to the given spatial distribution, and combining them with geometric structure-preserving numerical integrators which have good backward error properties, and preserve geometric invariants like the symplectic structure, energy, and momentum. Furthermore, these methods preserve the nonlinear structure of the configuration manifold, such as the Lie group or homogeneous space structure. The key technical goals include: (i) the development and analysis of structured integrators to accurately predict ergodic properties of a given system; (ii) the development of simulation-based optimization of system parameters, and controls to maximize efficiency of ergodic search; (iii) the generalization of these techniques to Lie groups and homogeneous spaces, enabling ergodic search for a rich class of robotic systems; (iv) experimental validation on realistic systems.These techniques will directly be applicable to a broad range of real-world industrial applications, including joint space exploration for robotic systems, fault detection in manufacturing, and optimal search, coverage, and information extraction for autonomous sensor networks. This industrial outreach will be facilitated by the release of public-domain software that will lower the barrier to adapting geometric numerical integration techniques in a variety of applications. Furthermore, we will engage in public outreach activities by teaming up with the Museum of Science and Industry in Chicago to develop an interactive exhibit demonstrating search algorithms for mechanical systems. These concrete applications serve to inspire high-school students (in particular, underrepresented minorities and women) to pursue STEM degrees, and this in turn will help to secure the long-term economic innovation and competitiveness of American industry.