Symmetries in Motion : Geometric Foundations of Motion

Some interesting aspects of motion and control for systems such as those found in biological and robotic locomotion, attitude control of spacecraft and underwater vehicles, and steering of cars and trailers, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can move forward or rotate in place. When the amplitude of the motion increases, the resulting net displacements normally increase as well. These observations lead to the general idea that when certain variables in a system move 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. Geometric tools that have been useful to describe this phenomenon are \connections", mathematical objects that are extensively used in general relativity and other parts of theoretical physics. The theory of connections, which is now part of the general subject of geometric mechanics, has also been helpful in the study of the stability or instability of a system and in its bifurcations under parameter variations. This approach, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in the attitude dynamics of spacecraft and underwater vehicles. The same theory also describes the behavior of systems with constraints, such as those found in a simple, non-slipping rolling wheel or for more complex systems like a car pulling many trailers or a snake sliding across a oor. The presence of symmetries in these systems, often exhibited as position and orientation invariance, leads to a general theory of reduction. In this theory, the salient features of the motion are highlighted in a manner that is also conducive to formulating control inputs. This article explains in a reasonably nontechnical way why some of these tools of geometric mechanics are useful in the study of motion control and locomotion generation.