In this paper we consider locomotion and manipulation using the notion of geometric phases as a central theme. Intuitively, geometric phases relate the motion of one parameter describing the configuration of a system to other parameters that undergo periodic motion. A simple example of geometric phase is the motion of an automobile performing a parallel parking maneuver. By moving the car backwards and forwards and turning the steering wheel in a periodic fashion, a driver is able to achieve a net sideways motion of the car even though the car cannot move sideways directly. This net sideways motion is the geometric phase associated with this choice of the car velocity and steering wheel angle.

The role of geometric phases as a means of analyzing locomotion is a relatively new perspective. One of the earliest works is that of Shapere and Wilczek (1989), who studied the motion of paramecia swimming in a highly viscous fluid. They show that periodic variations in the shape of an organism can be used to achieve net forward motion. This is very reminiscent of the type of motion present in parallel parking and this similarity can be made precise by using geometric phases.

There has also been an increased interest in the use of geometric phases for understanding motion in other biological systems, such as snakes and insects. Here again, periodic changes in one set of variables, which describe the shape of the system, are used to obtain net motion. The phasing of the inputs plays a central role, generating different gaits for achieving different types of motion. The interpretation of locomotion in terms of geometric phases is still far from complete, but it is providing a unifying view of locomotion and manipulation that has already yielded new insights and has impact on several challenging applications.

Locomotion involves movement of a mechanical system by appropriate application of forces on the robot. These forces can arise in several ways, depending on the means of locomotion used. The simplest form of locomotion is to apply the forces directly, as is done in a spacecraft, where high-energy mass is ejected in the direction opposite to the desired motion. A similar technique is the use of jet engines on modem aircraft.

For ground-based systems, a much more common means of locomotion is the use of forces of constraint between a robot and its environment. For example, a wheeled mobile robot exerts forces by applying a torque to its drive wheels. These wheels are touching the ground and, in the presence of sufficient friction, are constrained so as not to slip along the ground. This constraint is enforced by the application of internal forces, which cause a net force on the robot that propels it forward. If no constraints existed between the robot and the ground, then the robot would just spin its wheels. Similarly, for legged and snake robots, the parts of the robots in contact with the environment are used to exert net forces on the robot. In fact, for a large class of robotic systems we can view constraints as the basis for locomotion.

A second common feature in robot locomotion is the notion of base (or internal) variables versus fiber (or group) variables. Base variables describe the geometry and shape of the robot, while fiber variables describe its configuration relative to its environment. For example, in a snake robot the fiber variables might be the position and orientation of a coordinate frame fixed to the robot's body, while the base variables would be the angles that describe the overall shape of the robot. These base and fiber