Introduction

Geometry has been associated with motion, either implicitly or explicitly, from very early times in human history. There are relationships between motion and geometry both in how motion is described and in how it is harnessed and directed. Geometric notions underlie such mechanical devices as the potter's wheel and the wheeled cart, the ramp (or inclined plane), the lever, the pulley, and the coil. Although formal geometrical descriptions and explicit functionality principles were not supplied until centuries after such mechanisms came into widespread use, their connections with linked linear and circular motion, horizontal and vertical or forward and sideways motion, and winding-in and-out (spiral) or winding-up and-down (helical) motion are unmistakable. The substantial interrelationships between motion and geometry have been a continuing focus of scientific study and technological development from the eras of Archimedes of Syracuse, Leonardo da Vinci and Galileo Galilei, Rene Descartes, Isaac Newton, Pierre-Louis Moreau de Maupertuis, James Clerk Maxwell, and Albert Einstein fight through to the present time. Those linkages bear heavily on how motion is modeled and ultimately controlled, be it by mechanical contrivance (for instance, in a pendulum clock) or through the discovery of how prevailing conditions influence outcomes (for example, finding the trajectory of an object that is subject to gravity and that is thrown horizontally off a cliff).

From the construction of the Great Pyramids and of Stonehenge, which both involved the transport and careful positioning of massive blocks or lintels, to the reckoning of celestial motions; from the Renaissance design or engineering of a prototype submarine, bicycle, or helicopter to latter-day satellite positioning or in vivo intestinal exploration and examination; from the movements of subatomic particles to the meanderings of computer-modeled sidewinding snakes, geometry supplies an indispensable vocabulary for the mathematical description of whatever motions are observed, achievable, or sought. As mathematics is the language of science, so geometry is the language of motion. The motivation may have changed from a desire to understand, predict, or direct motions by way of geometric modeling and analysis to a focus on designing and controlling the mechanical generation of particular motions on the basis of their geometric description, computer simulation, and robotic replication. However, the value of this geometric language is undiminished.

Some of the modem developments described in the following chapters include the geometric control of robot motion and craft orientation, how high-power precision micromotors are engineered for less invasive surgery and self-focusing lens applications, what a mobile robot on a surface has in common with one moving in three dimensions, and how the motion-control problem is simplified by a coupled oscillator's geometric grouping of degrees of freedom and motion time scales.

The four papers in these proceedings provide a view through the scientific portal of today's motion-control geometric research into tomorrow's technology. The mathematics needed to carry out this research is that of modem differential geometry, and the questions raised in the field of motion-control geometry go directly to the research frontier. Some of the mathematical tools that are useful here are Lie algebras of vector fields, differential forms and exterior algebra, and affine connections. Another tool that has proven useful is gauge theory remarkably, the same sort of geometry that is used in elementary-particle physics. It is fortunate that mathematicians have developed the mathematical tools in a general context so that they can be used for many purposes. In particular, the mathematical notion of the holonomy of a connection has been around for some time—an idea that links locomotion generation



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Motion, Control, and Geometry: Proceedings of a Symposium Introduction Geometry has been associated with motion, either implicitly or explicitly, from very early times in human history. There are relationships between motion and geometry both in how motion is described and in how it is harnessed and directed. Geometric notions underlie such mechanical devices as the potter's wheel and the wheeled cart, the ramp (or inclined plane), the lever, the pulley, and the coil. Although formal geometrical descriptions and explicit functionality principles were not supplied until centuries after such mechanisms came into widespread use, their connections with linked linear and circular motion, horizontal and vertical or forward and sideways motion, and winding-in and-out (spiral) or winding-up and-down (helical) motion are unmistakable. The substantial interrelationships between motion and geometry have been a continuing focus of scientific study and technological development from the eras of Archimedes of Syracuse, Leonardo da Vinci and Galileo Galilei, Rene Descartes, Isaac Newton, Pierre-Louis Moreau de Maupertuis, James Clerk Maxwell, and Albert Einstein fight through to the present time. Those linkages bear heavily on how motion is modeled and ultimately controlled, be it by mechanical contrivance (for instance, in a pendulum clock) or through the discovery of how prevailing conditions influence outcomes (for example, finding the trajectory of an object that is subject to gravity and that is thrown horizontally off a cliff). From the construction of the Great Pyramids and of Stonehenge, which both involved the transport and careful positioning of massive blocks or lintels, to the reckoning of celestial motions; from the Renaissance design or engineering of a prototype submarine, bicycle, or helicopter to latter-day satellite positioning or in vivo intestinal exploration and examination; from the movements of subatomic particles to the meanderings of computer-modeled sidewinding snakes, geometry supplies an indispensable vocabulary for the mathematical description of whatever motions are observed, achievable, or sought. As mathematics is the language of science, so geometry is the language of motion. The motivation may have changed from a desire to understand, predict, or direct motions by way of geometric modeling and analysis to a focus on designing and controlling the mechanical generation of particular motions on the basis of their geometric description, computer simulation, and robotic replication. However, the value of this geometric language is undiminished. Some of the modem developments described in the following chapters include the geometric control of robot motion and craft orientation, how high-power precision micromotors are engineered for less invasive surgery and self-focusing lens applications, what a mobile robot on a surface has in common with one moving in three dimensions, and how the motion-control problem is simplified by a coupled oscillator's geometric grouping of degrees of freedom and motion time scales. The four papers in these proceedings provide a view through the scientific portal of today's motion-control geometric research into tomorrow's technology. The mathematics needed to carry out this research is that of modem differential geometry, and the questions raised in the field of motion-control geometry go directly to the research frontier. Some of the mathematical tools that are useful here are Lie algebras of vector fields, differential forms and exterior algebra, and affine connections. Another tool that has proven useful is gauge theory remarkably, the same sort of geometry that is used in elementary-particle physics. It is fortunate that mathematicians have developed the mathematical tools in a general context so that they can be used for many purposes. In particular, the mathematical notion of the holonomy of a connection has been around for some time—an idea that links locomotion generation

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Motion, Control, and Geometry: Proceedings of a Symposium with gauge theory. Interestingly, control and locomotion generation are two of the other areas in which these ideas can be applied. Geometry is a mathematical area too often neglected nowadays in a student's education. This publication will help adjust the control initially imposed about 2,300 years ago on one kind of "motion"— that of students entering Plato's Academy, where the following caveat was inscribed above the doorway: "Let no one ignorant of geometry enter here." Readers of these chapters will gain an appreciation of modem geometry and how it continues to play a crucial role in the context of motion control in cutting-edge science and technology.