Time-Optimal Control of One-Mode Flexible Structures: Analytical Solution and the Cost of Flexibility

This study investigates the time-optimal control for a double integrator coupled to a harmonic oscillator, providing the first analytical solution for rest-to-rest maneuvers. The research aims to derive a closed-form time-displacement law, which reveals that the synthesis of time-optimal control can be simplified to a single scalar inversion. The findings show that the maneuver time can be decomposed into a rigid-body minimum and a synchronization time, termed the cost of flexibility, demonstrating a scale-dependent penalty. For small maneuvers, the relationship indicates that one oscillator mode incurs a maneuver time cost equivalent to two additional integrators, while for larger maneuvers, flexibility's cost becomes asymptotically negligible. Furthermore, the analytical expressions indicate that adding flexibility to a rigid body never reduces maneuver time, although softening a flexible structure may enhance performance, challenging the notion that stiffer structures are always faster. The results underscore the significance of natural motions as robust design targets, with first-order sensitivity to stiffness disappearing in these conditions.