The use of tools from differential geometry to model and analyse a wide class of locomotion mechanisms in a unified way is explored. Robotic locomotion of the considered type is characterized by cyclic changes in the shape of a robot mechanism exploiting the constrained nature of a robot’s interaction with its environment to generate net motion. Examples for these types of robots are legged robots, snakelike robots, and wheeled mobile robots. The article provides a framework for understanding locomotion by concentrating on geometric structure. As explained in the introduction, the author’s aim is “to build a comprehensive theory for locomotion that is at once mathematically rigorous, elegant, and enlightening”.
In section 2 the most important used mathematical tools are explained. An introduction to principal bundles is given, and the relationship between locomotion and geometric phases is studied. The modeling of the constraint kinematics of a locomotion system in terms of a connection on a principal bundle is considered. The two-wheeled planar mobile robot is used as an introducing example.
In section 3 it is described how connections arise in the presence of Pfaffian constraints. It is shown how conservation laws arising from symmetries of mechanical systems may be described in terms of connections as well. Also other types of locomotion systems, for instance an inchworm robot, can be modeled, at least approximately, using connections on principal bundles.
In section 4 the controllability of systems is defined and analyzed on principal bundles. In the more general case, the geometric object of the curvature of a connection is used. Specific methods for generating motion through cyclic changes in the shape of a robot are discussed.
Finally, in the last section two longer examples illustrate the application of the presented ideas: side winding motion of a snake and tripod gait in a six-legged robot.