Vibration Analysis and Control of Magnet Positioner Using Curved-Beam Models

We develop a detailed dynamical model of a flexible robotic positioner which consists of two articulated permanent magnets and a suspended C-arm curved beam carrying them under gravitational and electromagnetic loads. The model is used to design a controller for positioning the magnets for medical navigation applications. An accurate nonlinear model of the structure is obtained as a combination of the rigid-body dynamics of the lumped masses and a finite-element model of the beam using straight-beam elements. These capture both in- and out-of-plane bending and twisting modes and permit large 3-D deformations. The fields and forces of the permanent magnets are approximated by dipoles. The static equilibrium of the structure and the dynamic linear approximation around it are computed. A reduced-order model which captures the first two natural modes is used for controller design. Using H-infinity theory, we design a controller for tracking smooth reference signals, disturbance attenuation, and stability robustness against uncertainties in material properties, the geometry, and neglected high-frequency modes. Simulations using the high-order model indicate the efficacy of this approach to vibration attenuation and tracking.