Dynamics, control and algorithms of rigid bodies and flexible structures

The problem of stabilization of rigid bodies and flexible structures has received a great deal of attention in recent years. People have developed a variety of feedback control laws to meet their design requirements and have formulated various but mostly open loop numerical algorithms for the dynamics of the corresponding closed loop systems.;Since the conserved quantities such as energy, momentum and symmetry play an important role in the dynamics, we investigate the conserved quantities for the closed loop control systems which formally or asymptotically stabilize rigid body rotation and modify the open loop numerical algorithms so that they preserve these important properties. The resulting closed loop algorithms are compared to standard techniques in several examples.;In the case of asymptotic stabilization of uniform motion of rigid bodies, it follows that the use of feedback introduces partial internal damping in the system dynamics. For such systems the dynamics converge to relative equilibria of the undamped system. In converging to these relative equilibria the dissipation introduced by the control law does work on the system and moves the spatially referenced angular momentum vector. We compare attitude drift for the closed loop dynamics with the undamped case and estimate the displacement of the spatially referenced angular momentum vector.;We further extend the problem of stabilization to the uniform rotation of a flexible rod. A three dimensional, geometrically exact rod model including shear, extension, torsion and flexure is stabilized by means of feedback torque applied to the boundary. The energy-momentum method of stability analysis is used as the basis of the feedback control design. We focus on the case of uniform axial rotation. We also present and formulate some algorithms for open loop, closed loop and dissipative geometrically exact rod systems. Several numerical examples are then compared with the linearized features.