precursors to the automated machines used for painting, welding, and pick-and-place operations in today's factories. For these types of automation and manufacturing tasks, the complexity of the robot can be minimized since the workspace of the robot can be carefully controlled and the robots are not required to perform particularly dextrous manipulation of objects. However, many future applications of robotics are moving toward more autonomous operation in highly uncertain environments. The robots being developed for these applications are increasingly complex and have a high degree of interaction with their environment.

Examples of the next generation of robots range from miniature robots for medical inspection and manipulation inside the human body to mobile robots for exploration in hazardous and remote environments. All these robots will require sensing, actuation, and computational capabilities that were unheard of just a few years ago. However, as we seek to design robots that can act with increasing autonomy, we move closer to endowing robots with human-like capabilities. And we begin to become limited by our own ability to understand and analyze the highly complex systems that we are trying to control.

As the complexity of robots increases, so does the importance of abstraction and theory in understanding and analyzing robot motion. One approach that has begun to yield new insights is the use of differential geometry, in the context of both geometric mechanics and nonlinear control theory. Two specific areas where progress is being made are locomotion and manipulation in robot systems.

Locomotion is defined as the act of moving from one place to another. For robots, there are several mechanisms by which this movement can occur. The use of wheels and of legs are the two traditional methods, but other possibilities, such as undulatory gaits in snake-like robots, have also been proposed and implemented. Each of these mechanisms has certain advantages over the others, but all of them fundamentally involve interaction with their environment. Locomotion is achieved by pushing, sliding, rolling, or a combination of all of these.

Robotic manipulation involves motion of an object rather than motion of the robot itself. The prototypical example is a multifingered hand manipulating a grasped object. Once again, the fundamental mechanisms that govern motion involve pushing, rolling, and sliding. The motion of a set of fingers grasping an object is constrained in much the same way as the motion of a legged robot is constrained by the contacts between its feet and the ground. Indeed, many of the tools that are used to analyze manipulation and grasping problems are easily adapted to analyze locomotion.

The most basic problem in all locomotion and manipulation systems is to devise a method for generating and controlling motion between one configuration and another. The common feature is that motion of the robot is constrained by its interactions with the environment. For example, in wheeled mobile robots the wheels must roll in the direction in which they are pointing and they must not slide sideways. In grasping, the motion of the fingers is constrained by the object being held in the grasp: motion of one finger affects the others since forces are transmitted between the fingers by the object. Even in legged robots, one usually assumes that the feet do not slip on the ground, allowing the robot to propel itself. These constraints on the motion of the system are the defining features for how locomotion and manipulation work in these systems.

Furthermore, in most locomotion and manipulation systems, the range of the actuators is small, while the desired net motion for the system may be large. A good example of this is using your fingers to screw in a light bulb: repeated grasping and twisting of the bulb is required in order to fully insert it into the socket. A large motion of the light bulb (multiple revolutions) is accomplished by repeated (i.e., periodic) small motions in your fingers.

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