Motion, Control, and Geometry Proceedings of a Symposium (1997) / Chapter Skim
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2: CYCLES THAT EFFECT CHANGE
2: CYCLES THAT EFFECT CHANGE
Pages 20-32
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From page 20... ...
The wheels of a steam locomotive should rotate steadily, yet their motive power comes from periodic processes involving the filling and emptying of the cylinders with steam and the corresponding motion of the pistons. In fact, both conventional piston-type steam engines and internal combustion engines depend on a carefully orchestrated, repetitive motion pattern as an intermediate step in the generation of steady rotational motion; the recently introduced vibratory motors depend on an even more subtle type of pattern generation.
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From page 21... ...
Theories dealing with nonlinear controllability provide considerable insight into the capabilities of systems of this latter type. Understanding the dynamics of their regulatory processes requires more study, and-only recently has there been an appropriate mathematical fo~-~ulation of a control problem in which pattern generation plays a decisive role.
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From page 22... ...
The power required to pump a fluid is the flow rate times the pressure. Applying these facts to the situation at hand, we see that, when expressed in suitable units, the energy supplied by the agent responsible for the flow u must provide energy at the rate u(x +y)
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From page 23... ...
For example, if the tanks are initially empty and if the bottom tank is filled to the level y = ~ while the top tank remains empty and then the top tank is filled to the level x = ~ while the bottom tank remains at y = I, then agent u supplies energy I/2 and agent v supplies energy 3/2. If we reverse the order, i.e., if agent v fills the top tank first and then agent u fills the bottom tank, it follows that the energy requirements of the agents are just reversed.
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From page 24... ...
To equate this to energy flow one needs a constant factor having suitable units. This factor can be expressed in terms of the value of a certain component of the Lie bracket of two vector fields; as u and v vary in a cyclical way, z increases monotonically with a rate proportional to the product of this factor and the area in (u,v)
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From page 25... ...
If we are told that a particular mark on the small wheel lies at 9 o'clock then we can say that a particular mark on the bigger wheel lies at one of two possible locations, separated by IS0 degrees. Again, the space of possible configurations of the bigger wheel forms a covering space for the set of configurations of the smaller one.
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From page 26... ...
If these modes were to vibrate at the same frequency and with the correct phase relationship, the overall motion of the tip protruding upward from the center would produce an elliptical path as shown at the bottom of the figure. Just as it takes two agents to manipulate the energy flow in the tank example, it takes two modes having the same frequency and appropriate phase relationship to generate the elliptical motion with nonzero area.
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From page 27... ...
~ __ ~ FIGURE 2.7 A multimode vibration of a beam. ~ _` __ ~ FIGURE 2.8 A mechanism for converting vibrational to translational motion.
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From page 28... ...
This process of conversion, called rectification, used to be a relatively inefficient and expensive process, requiring bulky equipment and producing unwanted heat. Today there exist solid state electronic devices capable of providing elegant and efficient solutions to this problem at modest cost.
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From page 29... ...
and v refer to the two switches and denote variables that represent the position of the switches in the diagram. The time evolution of this network is governed by a set of equations that are similar to those describing the tanks, although the meaning of the control terms is now different.
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From page 30... ...
In Figure 2.10 we diagram the relationship between the various causes and effects represented by the differential equations of our standard model. This type of block diagram differs from the usual physical diagrams, such as that shown in Figure 2.1, in that one does not attempt to be faithful to the physics or spatial relationships but, as happens when one represents natural phenomena in terms of equations, attempts to express the abstract relationships clearly.
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From page 31... ...
This can be interpreted as supporting the need for pattern generation. When one approaches the difficult problem of designing stabilizing feedback control laws for nonholonomic systems from this point of view, it often becomes much easier to understand.
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From page 32... ...
von Euler, K., 1985, "Central Pattern Generation During Breathing." In: The Motor System in Neurobiology, E Evarts, S
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