Geometric Phases, Control Theory, and Robotics

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Richard M. Murray
Proceedings of the Board on Mathematical Sciences, Science and Technology Symposium, Washington DC, 12 April 1994

Differential geometry and nonlinear control theory provide essential tools for studying motion generation in robot systems. Two areas where progress is being made are motion planning for mobile robots on the factory floor (or on the surface of Mars), and control of highly articulated robots---such as multifingered robot hands and robot ``snakes---for medical inspection and manipulation inside the gastrointestinal tract. A common feature of these systems is the role of constraints on the behavior of the system. Typically, these constraints force the instantaneous velocities of the system to lie in a restricted set of directions, but do not actually restrict the reachable configurations of the system. A familiar example in which this geometric structure can be exploited is parallel parking of an automobile, where periodic motion in the driving speed and steering angle can be used to achieve a net sideways motion. By studying the geometric nature of velocity constraints in a more general setting, it is possible to synthesize gaits for snake-like robots, generate parking and docking maneuvers for automated vehicles, and study the effects of rolling contacts on multifingered robot hands. As in parallel parking, rectification of periodic motions in the control variables plays a central role in the techniques which are used to generation motion in this broad class of robot systems.