2 6 O ct 1 99 9 10 / 12 / 99 MATHEMATICAL PROBLEMS IN THE CONTROL OF UNDERACTUATED SYSTEMS
نویسنده
چکیده
There are many interesting mathematical problems in control theory. In this paper we will discuss problems and techniques related to underactuated systems. An underactuated system is one with fewer control inputs than degrees of freedom. Balancing a ruler on the tip of a finger is a good example of an underactuated system. This system has five degrees of freedom (three for the fingertip and two angles for the ruler). However, only the three degrees of freedom for the fingertip are directly controlled. In fact, any system requiring balance is an underactuated system. A bicycle is an obvious example. An airplane is a less obvious example (six degrees of freedom, underactuated by two). We will give a mathematical formulation of several problems arising from applications, review some standard and new techniques, and pose some interesting and challenging open questions. Stabilization of underactuated systems To describe a mechanical system we start with a manifold, Q, representing all possible configurations of the system. The configuration space Q is equipped with a Riemannian metric, g, so that the kinetic energy is
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