Motion, Control, and Geometry Proceedings of a Symposium (1997) / Chapter Skim
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4: MOTION CONTROL AND COUPLED OSCILLATORS
4: MOTION CONTROL AND COUPLED OSCILLATORS
Pages 52-66
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From page 52... ...
4 Motion Control and Coupled Oscillators P.S. Krishnaprasad Department of Electrical Engineering and Institute for Systems Research University of Maryland ~:~ ~-~rol~Wres~of m~lex~te~gicat~: ~eE~.~:~:~:~:~:~ INTRODUCTION In optics, lithography applications in microelectronics, and in a variety of other contexts, the need for high-resolution motion control with high accuracy is met by specialized actuators that are quite different in their principles of operation from everyday devices such as electromagnetic motors.
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and armature A This actuator, consisting of three sleeves/tubes made from piezoelectric material mounted on a frame and enclosing a linear armature, works on the physical principle that the piezoelectric material deforms under electrical stimulus (the outer sleeves independently clamp down, and the middle sleeve stretches in length)
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It is precisely this mechanism, variously associated with geometric phases, area rules, and Lie bracket generation, that has had a crucial role as a tool for designing machines and algorithms to control them. In the context of motion generation, Brockett's paper (1989)
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In the section on "Unifying Geometry," we present a unifying geometric-mechanical picture of the ideas on rectification. The language of principal bundles and connections goes hand in hand with the mechanical notions of configuration spaces, symmetries, and constraints.
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, g= evolves in the group of rigid motions in the plane, with 'cost)
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(4.9) is the area of the loop in shape space executed by the unicyclist in the course of the chosen oscillatory maneuver.
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Shape change in that case is achieved via successively lifting and swinging the legs before resuming to ground contact. Clan 1n~f=.~.t mar Nathan=` `~r1th lash In ~1;= ~ ~ _: i_ `1 _ _1 ~ SCRIPTS AND OSCILLATORS Area rules of the type discussed in the last section have been used in developing computer programs to synthesize feedforward control laws (motion scripts)
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Global Motion , ~ Based on these insights from biology, one is led to a possible architecture for intelligent control of movement as in Figure 4.4. Here the command center communicates a prescribed global movement command to be transformed into symbolic and timing instructions, which are then implemented by a network of coupled oscillators that produce the script for shape change.
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are present then, it is possible to construct a well-defined splitting of the space of velocities (tangent bundle TQ of the configuration space) at every configuration, into a set of symmetry directions along group orbits (also called vertical directions)
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, leading to area rules. Once again the area rules yield constructive procedures for generating suitable movements in shape space to achieve required transport in configuration space.
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Apart from certain singular configurations, determined as those for which all three axles intersect (possibly at infinity) , the unifying geometry discussed previously applies and the "no sliding" constraints fix a principal connection.
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on the lamprey. The rich variety of global motions can be best understood by the proper synthesis of kinematic, geometric, and dynamic information, and the principle of rectification applied to cyclical shape variations is an effective guide even in this mathematically complex setting.
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Kopell, N., 1988, "Toward a Theory of Modelling Central Pattern Generators." In: Neutral Control of Rhythmic Movements in Vertebrates, A.H. Cohen, S
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{EKE Oceanic Engineering Society, 283-288, TEEE, New York. OF , Manikonda, V., 1994, A Hybrid Control Strategy for Path Planning and Obstacle Avoidance with Nonholonomic Robots, M.S.
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