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UBC Theses and Dissertations
Solution methods for differential systems subject to algebraic inequality constraints Spiteri, Raymond J.
Abstract
The method of programmed constraints has recently been proposed as an executable specification language for robot programming. The mathematical structures behind such problems are viability problems for control systems described by ordinary differential equations subject to user-defined inequality constraints. Although these types of problems are common in applications, practical algorithms and software are generally lacking. This thesis describes a new method for the numerical solution of such viability problems. The algorithm presented is composed of three parts: delay-free discretization, local planning, and local control. Delay-free discretizations are shown to be consistent discretizations of control systems described by ordinary differential equations with discontinuous inputs. The generalization of delay-free discretizations to higher order in the context of implicit-explicit Runge-Kutta methods represents a potentially powerful new class of time integrators for ordinary differential equations that contain terms requiring different discretizations. The local planning aspect is a computationally inexpensive way to increase the robustness in finding a solution to the viability problems of interest, making it a refinement to a strategy based on viability alone. The local control is based on the minimization of an artificial potential function in the form of a logarithmic barrier. Theoretical examples are given of situations where the choice of such a control can be interpreted to yield heavy solutions. Simulations of two robotic systems are then used to validate the particular strategy investigated. Some complementarity is shown between the programmedconstraint approach to robot programming and optimal control. Moreover, we demonstrate the relative efficiency of our algorithm compared to optimal control in the case of programming a mobile robot: our method is able to find a solution on the order of one hundred times faster than a typical optimal control solver. Some simulations on the control of a simple robot arm are also presented.
Item Metadata
Title |
Solution methods for differential systems subject to algebraic inequality constraints
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1997
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Description |
The method of programmed constraints has recently been proposed as an
executable specification language for robot programming. The mathematical structures
behind such problems are viability problems for control systems described by
ordinary differential equations subject to user-defined inequality constraints. Although
these types of problems are common in applications, practical algorithms
and software are generally lacking. This thesis describes a new method for the
numerical solution of such viability problems.
The algorithm presented is composed of three parts: delay-free discretization,
local planning, and local control. Delay-free discretizations are shown to be
consistent discretizations of control systems described by ordinary differential equations
with discontinuous inputs. The generalization of delay-free discretizations to
higher order in the context of implicit-explicit Runge-Kutta methods represents a
potentially powerful new class of time integrators for ordinary differential equations
that contain terms requiring different discretizations. The local planning aspect is
a computationally inexpensive way to increase the robustness in finding a solution
to the viability problems of interest, making it a refinement to a strategy based
on viability alone. The local control is based on the minimization of an artificial
potential function in the form of a logarithmic barrier. Theoretical examples are
given of situations where the choice of such a control can be interpreted to yield
heavy solutions.
Simulations of two robotic systems are then used to validate the particular
strategy investigated. Some complementarity is shown between the programmedconstraint
approach to robot programming and optimal control. Moreover, we
demonstrate the relative efficiency of our algorithm compared to optimal control in the case of programming a mobile robot: our method is able to find a solution on
the order of one hundred times faster than a typical optimal control solver. Some
simulations on the control of a simple robot arm are also presented.
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Extent |
6641443 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2009-04-17
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080007
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
1997-11
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.