Underactuated robots are characterized by a larger number of degrees of freedom than actuators and if they are designed with a specific mass distribution, they can be controlled by means of differential flatness theory. This structural property enables the development of lightweight and cost-effective robotic systems with enhanced dexterity. However, a key challenge lies in managing the passive joints, whose control demands precise and comprehensive dynamic modeling of the system. To simplify dynamic models, particularly for low-speed trajectories, friction is often neglected. While this assumption simplifies analysis and control design, it introduces residual oscillations of the end-effector about the target position. In this paper, the possibility of using optimal control along with differential flatness control is investigated to improve the tracking of the planned trajectories. First, the study was carried out through formal analysis, and then, it was validated by means of numerical simulations. Results highlight that optimal control can be used to plan the flat variables considering different (quadratic) performance indices: control effort, i.e. motor torque, and potential energy of the considered underactuated joint. Moreover, the minimization of potential energy can be used to design motion laws that are robust against variation of the stiffness and damping of the underactuated joint, thus reducing oscillations in the case of stiffness/damping mismatch.

翻译：欠驱动机器人具有自由度数目多于执行器数量的特征，若其质量分布经过特定设计，则可基于微分平坦理论实现控制。该结构特性使得开发具有增强灵巧性的轻量化、低成本机器人系统成为可能。然而，其核心挑战在于对被动关节的控制管理，这要求系统具备精确且完备的动力学建模。为简化动力学模型（尤其针对低速轨迹），摩擦效应常被忽略。虽然该假设简化了分析与控制设计，但会导致末端执行器在目标位置附近产生残余振荡。本文研究了结合最优控制与微分平坦控制以提升规划轨迹跟踪性能的可能性。首先通过理论分析开展研究，随后借助数值仿真进行验证。结果表明，最优控制可用于规划平坦变量，同时考虑不同（二次型）性能指标：控制能耗（即电机扭矩）以及所考察欠驱动关节的势能。此外，势能最小化可用于设计对欠驱动关节刚度和阻尼变化具有鲁棒性的运动规律，从而在刚度/阻尼失配情况下有效抑制振荡。