Coordinated path following: A nested invariant sets approach
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Date
2015-05-20
Authors
Doosthoseini, Alireza
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
In this thesis we study a coordinated path following problem for
multi-agent systems. Each agent is modelled by a smooth, nonlinear,
autonomous, deterministic control-affine ordinary differential
equation. Coordinated path following involves designing feedback
controllers that make each agent's output approach and traverse a
pre-assigned path while simultaneously coordinating its motion with
the other agents. Coordinated motion along paths includes tasks like
maintaining formations, traversing paths at a common speed and more
general tasks like making the positions of some agents obey functional
constraints that depend on the states of other agents.
The coordinated
path following problem is viewed as a nested set stabilization
problem. In the nested set stabilization approach, stabilization of
the larger set corresponds to driving the agents to their assigned
paths. This set, under suitable assumptions, is an embedded,
controlled invariant, product submanifold and is called the
multi-agent path following manifold. Stabilization of the nested set,
contained in the multi-agent path following manifold, corresponds to
meeting the coordination specification. Under appropriate assumptions,
this set is also an embedded controlled invariant submanifold which we
call the coordination set.
Our approach to locally solving nested set stabilization problems is
based on feedback equivalence of control systems. We propose and solve
two local feedback equivalence problems for nested invariant sets. The
first, less restrictive, solution gives necessary and sufficient
conditions for the dynamics of a system restricted to the larger
submanifold and transversal to the smaller submanifold to be linear
and controllable. This normal form facilitates designing controllers
that locally stabilize the coordination set relative to the
multi-agent path following manifold. The second, more restrictive,
result additionally imposes that the transversal dynamics to the
larger submanifold be linear and controllable. This result can
simplify designing controllers to locally stabilize the multi-agent
path following manifold. We propose sufficient conditions under which
these normal forms can be used to locally solve the nested set
stabilization problem.
To illustrate these ideas we consider a coordinated path following
problem for a multi-agent system of dynamic unicycles. The multi-agent
path following manifold is characterized for arbitrary paths. We show that each unicycle is feedback equivalent, in a
neighbourhood of its assigned path, to a system whose transversal and
tangential dynamics to the path following manifold are both double
integrators. We provide sufficient conditions under which the
coordination set is nonempty. The effectiveness of the proposed
approach is demonstrated experimentally on two robots.