Motion, Control, and Geometry Proceedings of a Symposium (1997) / Chapter Skim
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3: GEOMETRIC PHASES, CONTROL THEORY, AND ROBOTICS
3: GEOMETRIC PHASES, CONTROL THEORY, AND ROBOTICS
Pages 33-51
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From page 33... ...
They were a cross between numerically controlled milling machines and the master-slave teleoperators developed for handling radioactive material. These robots are the Note: Research supported in part by grants from the Powell Foundation, the National Science Foundation, and the National Aeronautics and Space Administration.
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From page 34... ...
As the complexity of robots increases, so does the importance of abstraction and theory in understanding and analyzing robot motion. One approach that has begun to yield new insights is the use of differential geometry, in the context of both geometric mechanics and nonlinear control theory.
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From page 35... ...
A simple example of geometric phase is the motion of an automobile performing a parallel parking maneuver. By moving the car backwards and forwards and turning the steering wheel in a periodic fashion, a driver is able to achieve a net sideways motion of the car even though the car cannot move sideways directly.
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From page 36... ...
Pfaffian constraints arise naturally in wheeled mobile robots: they model the ability of a wheel to roll along the ground and spin about its vertical axis but not slide sideways. Pfaffian constraints typically do not provide a complete model of the interaction with the environment, since frictional forces are present in both rolling and spinning, but they do capture the basic behavior of the system.
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From page 37... ...
. To test whether a set of vector fields are tangent to some surface, we make use of a special type of motion called a Lie bracket motion.
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From page 38... ...
323-324 for a derivation.) Motivated by this calculation, we define the Lie bracket of two vector fields go and g2 as Ego, g2 ~ = ~gq2 Gil - ~gq' g2 .
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From page 39... ...
The input vector fields are given by cosO O sins O if = tang g2 = 1 . We call go the drive vector field, corresponding to the motion commanded by the gas pedal, and g2 the steer vector field, corresponding to the motion ofthe steering wheel.' The Lie bracket between the drive and steer vector fields turns out to be g = g'g = _ 02 ~ O ~ We call g3 the wriggle vector field; it gives infinitesimal rotation of the car about the center of the rear wheels.
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From page 40... ...
The basic idea was to use sinusoids at integrally related frequencies to generate motion in the Lie bracket directions. In essence, one replaces the squares of a Lie bracket motion with circles.
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From page 41... ...
; and the open curve in x,y shows the increment in the y variable. The interesting implication here is that the Lie bracket motions correspond to rectification of harmonic periodic motions of the driving vector fields, and the harmonic relations are determined by the order of the Lie bracket corresponding to the desired direction of motion.
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From page 42... ...
The amount of motion in the fiber variables, due to a trajectory in the base variables, is the geometric phase associated with the path in the base space. Parallel parking corresponds to a phase shift in the y direction, while making a U-turn corresponds to a phase shift in the ~ direction (sometimes combined with y)
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From page 43... ...
A grasp of this type is called a manipulable grasp. For manipulable grasps, any object velocity x can be accommodated by some finger velocity vector ~ .
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From page 44... ...
If there are still sufficient degrees of freedom available for the fingers to roll on the object, then the internal motions parametenzed by u2 can be used to move the fingers individually around on the object. With the object position held fixed, each of the fingers can be controlled individually without regard to the motion of the others, simplifying the problem somewhat.
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From page 45... ...
If we treat "twisting" and "grasp/release" as inputs, then this type of motion corresponds exactly to a Lie bracket motion where the bracket direction corresponds to the motion of the bulb into the socket. The description of this problem does not quite fit into the differential framework described above without some modification of the underlying mathematics, but the basic notion of a Lie bracket motion is still present.
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From page 46... ...
, to be used to analyze the composition of Martian soil and rocks. The three primary goals of the MFEX rover mission are to complete a set of technology experiments in at least one soil type, complete an AXPS measurement on at least one rock with a video image of that rock, and take at least one full image of the lander.
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From page 47... ...
Medical Robotics: Minimally Invasive Surgery One of the exciting applications of robotic manipulation and locomotion is in medicine, particularly in minimally invasive surgery. Over the past ten years, the use of minimally invasive surgical techniques has increased dramatically in the U.S.
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From page 48... ...
However, with new techniques and new medical instruments, the use of minimally invasive techniques has surged, and this is by far the most common method currently in use for gall bladder removal. Another minimally invasive surgical technique is the use of an endoscope for inspection and removal of tissue or polyps from the gastrointestinal tract.
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From page 49... ...
Different gaits can be achieved in the snakeboard by using integrally related periodic motions in the input variables of the system. As its name indicates, the snakeboard provides an important link between wheeled mobile robots and more complicated snake-like robots.
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From page 50... ...
and S.S. Sastry, 1993, "Nonholonomic Motion Planning: Steering Using Sinusoids," IEEE Transactions on Automatic Control 38(5)
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From page 51... ...
and G Tang, 1991, "Shortest Paths for the Reeds-Shepp Car: A Worked out Example of the Use of Geometric Techniques in Nonlinear Optimal Control," Technical Report, New Brunswick, NI: Rutgers Center for Systems and Control.
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