Continuum soft robots are inherently underactuated and subject to intrinsic input constraints, making dynamic control particularly challenging, especially in hybrid rigid-soft robots. While most existing methods focus on quasi-static behaviors, dynamic tasks such as swing-up require accurate exploitation of continuum dynamics. This has led to studies on simple low-order template systems that often fail to capture the complexity of real continuum deformations. Model-based optimal control offers a systematic solution; however, its application to rigid-soft robots is often limited by the computational cost and inaccuracy of numerical differentiation for high-dimensional models. Building on recent advances in the Geometric Variable Strain model that enable analytical derivatives, this work investigates three optimal control strategies for underactuated soft systems-Direct Collocation, Differential Dynamic Programming, and Nonlinear Model Predictive Control-to perform dynamic swing-up tasks. To address stiff continuum dynamics and constrained actuation, implicit integration schemes and warm-start strategies are employed to improve numerical robustness and computational efficiency. The methods are evaluated in simulation on three Rigid-Soft and high-order soft benchmark systems-the Soft Cart-Pole, the Soft Pendubot, and the Soft Furuta Pendulum- highlighting their performance and computational trade-offs.

翻译：连续体软体机器人本质上是欠驱动的，并受到固有的输入约束，这使得动态控制尤为困难，尤其是在混合刚性-柔性机器人中。虽然现有方法大多关注准静态行为，但诸如摆起之类的动态任务需要精确利用连续体动力学。这导致了对简单低阶模板系统的研究，但这些系统往往无法捕捉真实连续体变形的复杂性。基于模型的最优控制提供了一种系统性的解决方案；然而，其在刚性-柔性机器人上的应用常常受限于高维模型数值微分的计算成本和精度不足。基于近期在几何可变应变模型（该模型能够实现解析导数）方面的进展，本研究探讨了三种用于欠驱动软体系统的最优控制策略——直接配点法、微分动态规划和非线性模型预测控制——以执行动态摆起任务。为应对刚性的连续体动力学和受限的驱动问题，采用了隐式积分方案和热启动策略，以提高数值鲁棒性和计算效率。这些方法在三个刚性-柔性及高阶软体基准系统——软体小车摆、软体双摆和软体古村摆——上进行了仿真评估，突出了它们的性能表现和计算权衡。