Collaborative Research: Geometrically Optimal Gait Optimization

动物的运动很难从第一原理精确建模，理想化的数学近似往往忽略了潜在的重要物理效应。然而，这些数学描述是理解自然运动的最有力工具，也是在工程机器人系统中复制其有效性的最有力工具。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，用于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（例如行进效率）的运动，但需要动力学的全面数学模型。另一方面，数据驱动的Floquet分析（DDFA）允许基于观察的重复运动的动力学建模，但仅提供系统行为的狭窄画像。本计画将应用DDFA来建构几何模型，进而利用几何控制的方法来寻找所需的步态。通过从系统的物理过程的观察输出建立运动模型，该项目将允许真实的运动的复杂性被容纳到强大的几何设计框架中，有效地使用测量。研究结果将通过使机器人能够强壮有效地行走，游泳或爬行，以促进国家的繁荣和福利，用于搜救或环境监测等任务。该项目还将深入了解动物的运动策略。该项目包括一个学生外展部分，提供有关步态的动手学习模块。该项目结合了两种范式，从严格的数学角度考虑系统与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，着眼于如何在整个配置空间的系统动力学变化。这种全局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Floquet理论，认为步态周期是固定的，并分析扰动远离其周期性运动。从这个角度来看，步态是一组身体形状和速度的耦合振荡。这部分工作试图通过经验观察来理解耦合的性质，但只提供了固定步态附近的系统动力学的局部视图，并且没有提供明确的矢量沿着来优化这些步态。该项目将几何和数据驱动的Floquet范式统一起来，结合了它们的优势，同时减轻了它们的弱点。它将最优性的几何概念引入Floquet分析，并将Floquet范式中的数据驱动建模技术引入几何建模方法。在具有不同人体拓扑结构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，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.