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：欠驱动机器人具有执行器数量少于自由度的特性，通过特定质量分布设计可借助微分平坦理论实现控制。该结构特性使得开发轻量化、低成本且具有更高灵活性的机器人系统成为可能。然而关键挑战在于被动关节的管理，其控制需要对系统进行精确且全面的动力学建模。为简化动力学模型（特别是针对低速轨迹），常忽略摩擦力。该假设虽简化了分析与控制设计，却会导致末端执行器在目标位置附近产生残余振荡。本文研究了将最优控制与微分平坦控制相结合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，随后通过数值仿真进行验证。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：控制能量（即电机力矩）与所考虑欠驱动关节的势能。此外，势能最小化可用于设计对欠驱动关节刚度与阻尼变化具有鲁棒性的运动规律，从而在刚度/阻尼失配情况下减少振荡。