Collaborative Research: Geometrically Optimal Gait Optimization

动物的运动很难从基本原理中准确地建模，理想化的数学近似往往忽略了潜在的重大物理影响。然而，这些数学描述是理解自然运动并在工程机器人系统中复制其有效性的最强大的工具。该项目将强大的数学分析和设计框架与数据驱动的方法相结合，以开发反映观察到的行为的预测关系。几何控制方法允许构造优化某些有益属性的运动，例如旅行效率，但需要全面的动力学数学模型。另一方面，数据驱动的Floquet分析(DDFA)允许基于观测对重复运动的动力学进行建模，但仅提供系统行为的狭隘描述。该项目将应用DDFA来构建几何模型，然后使用几何控制的方法来找到所需的步态。通过从系统物理过程的观测输出建立运动模型，该项目将允许将实际运动的复杂性纳入强大的几何设计框架，并有效地使用测量。这一成果将使机器人能够健壮高效地行走、游泳或爬行，执行搜救或环境监测等任务，从而促进国家的繁荣和福祉。该项目还将使人们深入了解动物的运动策略。该项目包括一个学生推广部分，其中包括提供有关步态的实践学习的模块。这个项目结合了两个范例，从严格的数学角度考虑系统和环境之间的全身相互作用。一种方法基于规范理论和几何力学，研究系统动力学如何在配置空间中变化。这种全球视角允许定义最佳步态并研究其特征，但依赖于详细的系统模型。第二种方法植根于弗洛奎特理论，将步态周期视为固定的，并分析远离其周期运动的扰动。从这个角度来看，步态是一组身体形状和速度的耦合振荡。这项工作试图通过经验观察来理解耦合的本质，但只提供了固定步态附近系统动力学的局部视图，并没有提供优化这些步态的明确矢量。该项目统一了几何和数据驱动的Floquet范例，结合了它们的优势，同时减轻了它们的弱点。它将最优化的几何概念引入FLOQUET分析，并将数据驱动的建模技术从FLOQUET范例引入几何建模方法。在具有不同身体拓扑和环境相互作用的一系列系统上进行的实验将在这一新框架的开发和评估中发挥关键作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Animal locomotion is difficult to model accurately from first principles, and idealized mathematical approximations often neglect potentially significant physical effects. Yet these mathematical descriptions are the most powerful tools available to understand natural movement, and to replicate its effectiveness in engineered robotic systems. This project combines a powerful mathematical analysis and design framework with a data-driven method for developing predictive relations that reflect observed behavior. The geometric control approach allows the construction of motions that optimize certain beneficial attributes, such as the efficiency of travel, but requires comprehensive mathematical models of the dynamics. On the other hand, Data-Driven Floquet Analysis (DDFA) allows modeling of the dynamics of repetitive motions based on observations, but provides only a narrow portrait of the system behavior. This project will apply DDFA to construct geometric models, which will then enable use of the methods of geometric control to find desirable gaits. By building locomotion models from the observed outputs of the system's physical processes, this project will allow the complexities of real motions to be accommodated into powerful geometric design frameworks, with an efficient use of measurements. The results will advance the nation's prosperity and welfare by enabling robots that walk, swim, or crawl robustly and efficiently, for missions such as search-and-rescue or environmental monitoring. The project will also give insight on the locomotion strategy of animals. The project includes a student outreach component, with modules that provide hands-on learning about gaits. This project combines two paradigms which consider whole-body interaction between a system and its environment from a rigorous mathematical perspective. One approach, based on gauge theory and geometric mechanics, looks at how the system dynamics vary across the configuration space. This global perspective allows for optimal gaits to be defined and their characteristics studied, but relies on detailed system models. The second approach, rooted in Floquet theory, views the gait cycle as fixed and analyzes perturbations away from its cyclic motions. In this perspective, a gait is a set of coupled oscillations in body shape and velocity. This body of work seeks to understand the nature of the coupling through empirical observation, but provides only local views of the system dynamics near fixed gaits, and does not provide clear vectors along which to optimize those gaits. This project unifies the geometric and data-driven Floquet paradigms in a way that combines their strengths while mitigating their weaknesses. It brings geometric notions of optimality into Floquet analysis and data-driven modeling techniques from the Floquet paradigm into the geometric modeling approach. Experiments on a range of systems with different body topologies and environmental interactions will play a key role in both the development and evaluation of this new framework.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.