AI 中文总结
该研究针对耦合简谐振子的双积分器静止到静止机动，推导出时间最优控制的首个解析解，揭示了柔性对机动时间的尺度依赖代价及非单调刚度关系，给出鲁棒设计目标。
AI 中文摘要
双积分器的时间最优解是一个基础性结果：其归一化步长L对应的时间-位移定律为T_r=2√L，这是数值方法无法提供的结构。我们求解了耦合到简谐振子的双积分器从静止到静止机动的时间最优控制，推导出首个解析解和闭式时间-位移定律，综合过程简化为单个标量求逆。该模型可简化为两质量弹簧系统、单弯曲模态柔性结构和线性化桥式起重机。毕达哥拉斯恒等式T²=T_r²+2T_s²将最优机动时间分解为刚体最小值和同步时间T_s≤π（即柔性的代价），给出尖锐包络2√L≤T≤2√(L+π²/2)。该代价与尺度相关：小机动时T∝L^(1/4)，一个振子模态的代价相当于两个额外积分器；大机动时柔性渐近自由，在振子完成整周期的自然运动时代价消失。闭式灵敏度公式证明机动时间对刚度非单调：向刚体添加柔性永远不会缩短机动时间，但软化已有的柔性结构可以做到，即刚度更高并非总是更快。自然运动是鲁棒设计目标，因为此处对刚度的一阶灵敏度消失。
英文摘要
The time-optimal solution for a double integrator is a foundational result: its time-displacement law $T_r = 2\sqrt{L}$ for a normalized step $L$ is structure that numerical methods cannot provide. We solve the time-optimal control for rest-to-rest maneuvers of a double integrator coupled to a harmonic oscillator, deriving the first analytical solution and closed-form time-displacement law. Synthesis reduces to a single scalar inversion. This is the model to which the two-mass-spring system, the one-bending-mode flexible structure, and the linearized overhead crane reduce. A Pythagorean identity $T^2 = T_r^2 + 2T_s^2$ decomposes the optimal maneuver time into the rigid-body minimum and a synchronization time $T_s \leq π$, the cost of flexibility, giving the sharp envelope $2\sqrt{L} \leq T \leq 2\sqrt{L + π^2/2}$. The penalty is scale-dependent. For small maneuvers $T \propto L^{1/4}$: one oscillator mode costs as much as two additional integrators. For large ones, flexibility is asymptotically free. It vanishes at the natural motions, where the oscillator completes whole cycles. A closed-form sensitivity formula proves the maneuver time non-monotone in stiffness. Adding flexibility to a rigid body never shortens a maneuver, yet softening an already flexible structure can: stiffer is not always faster. The natural motions are robust design targets since first-order sensitivity to stiffness vanishes there.
