**Original manuscript:** 2009/07/09

In this thesis is initiated a more systematic geometric exploration of energy
shaping. Most of the previous results have been dealt with particular cases
and neither the existence nor the space of solutions has been discussed with
any degree of generality. The geometric theory of partial differential
equations originated by Goldschmidt and Spencer in late 1960s is utilized to
analyze the partial differential equations in energy shaping. The energy
shaping partial differential equations are described as a fibered submanifold
of a *k*-jet bundle of a fibered manifold. By revealing the nature of
kinetic energy shaping, similarities are noticed between the problem of
kinetic energy shaping and some well-known problems in Riemannian geometry.
In particular, there is a strong similarity between kinetic energy shaping
and the problem of finding a metric connection initiated by Eisenhart and
Veblen. We notice that the necessary conditions for the set of so-called
lambda-equation restricted to the control distribution are related to the
Ricci identity, similarly to the Eisenhart and Veblen metric connection
problem. Finally, the set of lambda-equations for kinetic energy shaping are
coupled with the integrability results of potential energy shaping. This
gives new insights for answering some key questions in energy shaping that
have not been addressed to this point. The procedure shows how a poor design
of closed-loop metric feedback can make it impossible to achieve any
flexibility in the character of the possible closed-loop potential function.
The integrability results of this thesis have been used to answer some
interesting questions about the energy shaping method. In particular, a
geometric proof is provided which shows that linear controllability is
sufficient for energy shaping of linear simple mechanical systems.
Furthermore, it is shown that all linearly controllable simple mechanical
control systems with one degree of underactuation can be stabilized using
energy shaping feedback. The result is geometric and completely
characterizes the energy shaping problem for these systems. Using the
geometric approach of this thesis, some new open problems in energy shaping
are formulated. In particular, we give ideas for relating the kinetic energy
shaping problem to a problem on holonomy groups. Moreover, we suggest that
the so-called Fakras lemma might be used for investigating the stabilization
condition of energy shaping.

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