Dynamical systems (DSs) provide a framework for high flexibility, robustness, and control reliability and are widely used in motion planning and physical human-robot interaction. The properties of the DS directly determine the robot's specific motion patterns and the performance of the closed-loop control system. In this paper, we establish a quantitative relationship between stiffness properties and DS. We propose a stiffness encoding framework to modulate DS properties by embedding specific stiffnesses. In particular, from the perspective of the closed-loop control system's passivity, a conservative DS is learned by encoding a conservative stiffness. The generated DS has a symmetric attraction behavior and a variable stiffness profile. The proposed method is applicable to demonstration trajectories belonging to different manifolds and types (e.g., closed and self-intersecting trajectories), and the closed-loop control system is always guaranteed to be passive in different cases. For controllers tracking the general DS, the passivity of the system needs to be guaranteed by the energy tank. We further propose a generic vector field decomposition strategy based on conservative stiffness, which effectively slows down the decay rate of energy in the energy tank and improves the stability margin of the control system. Finally, a series of simulations in various scenarios and experiments on planar and curved motion tasks demonstrate the validity of our theory and methodology.

翻译：动力系统（DSs）为高灵活性、鲁棒性和控制可靠性提供了框架，广泛应用于运动规划和人机物理交互中。动力系统的特性直接决定了机器人的具体运动模式以及闭环控制系统的性能。本文建立了刚度特性与动力系统之间的定量关系。我们提出了一种刚度编码框架，通过嵌入特定刚度来调制动力系统的特性。特别地，从闭环控制系统无源性的角度出发，通过编码保守刚度来学习保守动力系统。所生成的动力系统具有对称吸引行为和可变刚度分布。所提方法适用于属于不同流形和类型（如闭合和自相交轨迹）的示教轨迹，且在不同情况下始终保证闭环控制系统具有无源性。对于跟踪一般动力系统的控制器，系统的无源性需要通过能量库来保证。我们进一步提出了一种基于保守刚度的通用矢量场分解策略，该策略有效减缓了能量库中能量的衰减速率，提高了控制系统的稳定裕度。最后，通过多种场景下的系列仿真以及在平面和曲面运动任务上的实验，验证了我们理论与方法的有效性。