CAREER: Geometric Understanding of Locomotion

这个学院早期职业发展（Career）奖将创建一个严格的数学框架，用于分析动物在世界上推动自己的方式，目标是设计接近并超越自然系统能力的仿生机器人。例如，了解蛇鳞的摩擦特性如何帮助动物在泥泞中移动，可以为危险地形中的蛇形机器人系统设计智能皮肤和推进步态。特别是，该项目将使用强大的几何技术来研究目前知之甚少的系统的运动，如柔性和连续变形的软机器人，或系统与地面间歇性接触的运动。这一研究成果将极大地推动创新型机器人尤其是软机器人的设计。这项工作的更广泛的影响将包括几何方法在非数学家的更广泛社区中的渗透，通过动物运动的可访问示例，以及教科书和随附的可视化软件来促进。当系统有关节限制时（例如，它们有四肢而不是轮子或螺旋桨），它们的运动能力取决于它们在步态周期的不同点改变与环境相互作用的效率。当这些相互作用是一阶约束并且它们随系统形状平滑变化时，机车有效性可以通过李氏支架（与矢量场旋度密切相关的结构）来表征。该项目试图将这一概念扩展到包括方向依赖效应（例如，来自向后指向的刺或鬃毛的摩擦），二阶动力学（例如，弹性尾巴或空气中的翅膀），无限维系统和混合系统（例如，可以将脚从地面抬起的步行者）。该项目将使几何分析可访问的具体系统包括：1)具有许多形状变量的系统，其曲率是高维结构；2)混合动力系统，其动态曲率有“角”；3)棘轮系统，其反作用力取决于相对运动的符号；4)弹性系统，其步态周期部分来自被动动力学；5)滑行系统，其步态有效性更好地表征动量传递而不是位移诱导。

This Faculty Early Career Development (CAREER) award will create a rigorous mathematical framework for analysis of the ways in which animals propel themselves through the world, with the goal of designing bio-inspired robots that approach and surpass the capabilities of natural systems. For example, understanding how the frictional properties of snake scales helps the animal move through mud, could lead to the design of smart skins and propulsive gaits for snake-like robotic systems in hazardous terrain. In particular, this project will use powerful geometric techniques to study the locomotion of systems that are currently poorly understood, like flexible and continuously deformable soft robots, or of movements in which the system makes intermittent contact with the ground. The outcomes of this research will greatly advance the design of innovative robots, especially soft robots. Broader impacts of this work will include increased penetration of geometric methods into the broader community of non-mathematicians, facilitated by accessible examples from animal locomotion, and by a textbook and accompanying visualization software.When systems have joint limits (i.e., they have limbs instead of wheels or propellers), their ability to locomote depends on how effectively they can change their interaction with the environments at different points in a gait cycle. When these interactions are first-order constraints and they change smoothly with the system's shape, locomotive effectiveness can be characterized via a Lie bracket (a structure closely related to the curl of a vector field). This project seeks to extend this concept to include direction-dependent effects (e.g., friction from backwards-pointing spines or bristles), second order dynamics (e.g., elastic tails or wings in air), infinite-dimensional systems, and hybrid systems (e.g. walkers that can lift their feet from the ground. Specific systems that will be made accessible to geometric analysis by this project include, 1) systems with many shape variables, whose curvature is a high-dimensional structure; 2) hybrid systems, which have "corners" in their dynamic curvature; 3) ratcheting systems, whose reaction forces depend on the sign of the relative motion; 4) elastic systems, whose gait cycles partially emerge from their passive dynamics; and 5) gliding systems, whose gait effectiveness is better characterized by momentum transfer than by displacement induced.