Geometric Foundations of Motion and Control

Some interesting aspects of motion and control such as those found in biological and robotic locomotion, and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the general idea that when one variable in a system moves in a periodic fashion, motion of the whole object can result. This property can be used for control purposes; the position and attitude of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. One of the geometric tools that has been used to describe this phenomenon is that of connections, a notion that is extensively used in general relativity and other parts of theoretical physics. This tool, part of the general subject of geometric mechanics, has been helpful in the study of the stability or instability of a system and in its bifurcations, that is, changes in the nature of the systems dynamics, as some parameter changes. Geometric mechanics, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in attitude dynamics. The theory is also being developed for systems with rolling constraints such as those found in a simple rolling wheel. This article explains how some of these tools of geometric mechanics are used in the study of motion control and locomotion generation.