The Geometry and Control of Dissipative Systems

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Scott D. Kelly and Richard M. Murray
1996 IEEE Control and Decision Conference

We regard the internal configuration of a deformable body, together with its position and orientation in ambient space, as a point in a trivial principal fiber bundle over the manifold of body deformations. In the presence of a symmetry which leads to a conservation law, the self-propulsion of such a body due to cyclic changes in shape is described by the corresponding mechanical connection on the configuration bundle. In the presence of viscous drag sufficient to negate inertial effects, the viscous connection takes the place of the mechanical connection. Both connections may be represented locally in terms of the variables describing the body's shape. In the presence of both inertial and viscous effects, the equations of motion may be written in terms of the two local connection forms as an affine control system with drift on the manifold of configurations and body momenta. We apply techniques from nonlinear control theory to the equations in this form to obtain criteria for a particular form of accessibility.