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Aspects of Geometric Mechanics and Control of Mechanical Systems

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Andrew D. Lewis

PhD Dissertation, Caltech, Jun 1995

any interesting control systems are mechanical control systems. In spite of
this, there has not been much effort to develop methods which use the special
structure of mechanical systems to obtain analysis tools which are
suitable for these systems. In this dissertation we take the first steps
towards a methodical treatment of mechanical control systems.

First we develop a framework for analysis of certain classes of
mechanical control systems. In the Lagrangian formulation we study ``simple
mechanical control systems'' whose Lagrangian is ``kinetic energy minus
potential energy.'' We propose a new and useful definition of
controllability for these systems and obtain a computable set of conditions
for this new version of controllability. We also obtain decompositions of
simple mechanical systems in the case when they are not controllable. In the
Hamiltonian formulation we study systems whose control vector fields are
Hamiltonian. We obtain decompositions which describe the controllable and
uncontrollable dynamics. In each case, the dynamics are shown to be
Hamiltonian in a suitably general sense.

Next we develop intrinsic descriptions of Lagrangian and Hamiltonian
mechanics in the presence of external inputs. This development is a first
step towards a control theory for general Lagrangian and Hamiltonian
control systems. Systems with constraints are also studied. We first give a
thorough overview of variational methods including a comparison of the
``nonholonomic'' and ``vakonomic'' methods. We also give a generalised
definition for a constraint and, with this more general definition, we are
able to give some preliminary controllability results for constrained systems.

CDS Technical Report
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Richard Murray (murray@cds.caltech.edu)
Last modified: Tue Aug 30 07:42:21 2005