CAREER: Geometric Understanding of Locomotion

点击翻译按钮获取中文摘要

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.