Obstructions to Motion Planning by the Continuation Method
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Authors
Amiss, David Scott Cameron
Date
2013-01-03
Type
thesis
Language
eng
Keyword
geometric control theory , motion planning , continuation method , path-lifting equations , singular controls , nonlinear control theory
Alternative Title
Abstract
The subject of this thesis is the motion planning algorithm known as the continuation method. To solve motion planning problems, the continuation method proceeds by lifting curves in state space to curves in control space; the lifted curves are the solutions of special initial value problems called
path-lifting equations. To validate this procedure, three distinct obstructions
must be overcome. The first obstruction is that the endpoint maps of the control system
under study must be twice continuously differentiable. By extending a result
of A. Margheri, we show that this differentiability property is satisfied by an
inclusive class of time-varying fully nonlinear control systems. The second obstruction is the existence of singular controls, which are simply the singular points of a fixed endpoint map. Rather than attempting to completely characterize such controls, we demonstrate how to isolate control systems for which no controls are singular. To this end, we build on the
work of S. A. Vakhrameev to obtain a necessary and sufficient condition. In particular, this result accommodates time-varying fully nonlinear control
systems. The final obstruction is that the solutions of path-lifting equations may not
exist globally. To study this problem, we work under the standing assumption
that the control system under study is control-affine. By extending a result of Y. Chitour, we show that the question of global existence can be resolved by examining Lie bracket configurations and momentum functions. Finally, we show that if the control system under study is completely
unobstructed with respect to a fixed motion planning problem, then its corresponding endpoint map is a fiber bundle. In this sense, we obtain a necessary condition for unobstructed motion planning by the continuation method.
Description
Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2012-12-18 20:53:43.272
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