Motion, Control, and Geometry Proceedings of a Symposium (1997) / Chapter Skim
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1: GEOMETRIC FOUNDATIONS OF MOTION AND CONTROL
1: GEOMETRIC FOUNDATIONS OF MOTION AND CONTROL
Pages 3-19
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From page 3... ...
Perhaps the most popular example of the generation of rotational motion is the falling cat, which is able to execute a 180° reorientation, all the while having zero angular momentum. It achieves this by manipulating its joints to create shape changes.
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Phase formulas for special problems such as rigid body motion and polarized light in helical fibers were understood already in the early 1950s. Additional historical comments and references can be found in Berry (1990)
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FIGURE 1.2 Rolling your finger in a circular motion on a rolling sphere generates rotations.
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One can generate translational motion as well as rotational motion. For example, microorganisms and snakes generate translations by a very specific cyclic manipulation of their internal variables (Shapere and Wilczek, 1987)
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geometric phase vertical direction horizontal directions JO -\' -- - /1 : ~ a' 1/ ~ bundle projection base space FIGURE 1.3 A connection divides the space into vertical and horizontal directions. bundle In general, we can expect that if we have a horizontal motion in the bundle and if the corresponding motion in the base is cyclic, then the horizontal motion will undergo a shift, which we will call a phase shift, between the beginning and the end of its path.
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This is one reason that areas so commonly appear in geometric phase formulas. Connections are ubiquitous in geometry and physics.
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-- a ~ -- ~ -rr ~ "7 The planar skater illustrates well some of the basic ideas of geometric phases. If the device starts with zero angular momentum and it moves its arms in a periodic fashion, then the whole assemblage can rotate, keeping, of course, zero angular.
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Let us briefly indicate how geometric phases come into the rigid body example. Suppose we are given a trajectory U(t)
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Geometrically we can picture the rigid body as tracing out a path in its phase space; that is, the space of rotations (playing the role of positions) and corresponding momenta with the constraint of a fixed value of the spatial angular momentum.
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dynamic phased geometric phase / spherical cap solid angle I' , true trajectory /horizontal lift using We mechanical connection FIGURE 1.6 The geometric phase formula for rigid body motion. map to body variables Reduced trajectory mometum sphere This formula for the rigid body phase has a long and interesting history.
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From page 13... ...
This is already well illustrated by a toy called the rattleback, a canoe-shaped piece of wood or plastic. When the rattleback rocks on a flat surface like a table, the rocking motion induces a rotational motion, so that it can go from zero to nonzero angular momentum about the vertical axis as a result of the interaction of the rocking and rotational motion and the rolling constraint with the table.
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From page 14... ...
The situation is really not much different from what people do everyday when they ride a bicycle. One of the interesting things is that the subjects that have come before-namely, the use of connections in stability theory-can be turned around to be used to find useful stabilizing controls, for example, how to control the onboard gyroscopes in a spacecraft to stabilize the otherwise unstable motion about the middle axis of a rigid body (see Bloch et al., 1992; Kammer and Gray, 1993)
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In the context of problems like the falling cat, a remarkable consequence of the Maximum Principle is that, relative to an appropriate cost Unction, the optimal trajectory in the base space is a trajectory of a Yang-Mills particle. The equations for a Yang-Mills particle are a generalization of the classical Lorentz equations for a particle with charge e in a magnetic field B: d~v=-vxB, where v is the velocity of the particle and where c is the velocity of light.
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With this set-up, one is now in a good position to identify the resulting geometric phase with the holonomy of a connection that is a synthesis of the kinematic and mechanical connection. Carrying this out and implementing these ideas for more complex systems is in fact the subject of current research.
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From page 17... ...
Berry, M., 1990, "Anticipations of the geometric phase," Physics Todlay, December, 34- 40. Berry, M., 1985, "Classical adiabatic angles and quantal adiabatic phase,"~J.
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Krishnaprasad, and J.E. Marsden, 1989, "The Dynamics of Coupled Planar Rigid Bodies Part 2: Bifurcations, Periodic Solutions, and Chaos," Dynamics and Differential Equations 1, 269-298.
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From page 19... ...
and S Sastry, 1995, "On reorienting linked rigid bodies using internal motions," IEEE Trans.
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