Whenever a robotic task needs to be defined and adapted based on a reference motion, Dynamic Movement Primitives (DMP) represent a standard and efficient method for encoding it. The nominal trajectory is typically obtained through a Programming by Demonstration (PbD) approach, where the robot is taught a specific task through kinesthetic guidance. Subsequently, the motion is reproduced by the manipulator in terms of both geometric path and timing law. The basic approach for modifying the duration of the execution involves adjusting a time constant characterizing the model. On the contrary, the goal of this paper is to achieve a complete decoupling between the geometric information of the task and the timing law governing the execution, thanks to a new spatial sampling algorithm. This leads to a new DMP concept called Geometric DMP (GDMP), which exhibits the property of being phase-free since the phase variable is no longer constrained to the demonstration timing law. GDMP open up to a variety of applications, including task duration optimization subject to velocity and acceleration constraints and human-in-the-loop applications in co-manipulation tasks. With reference to the latter application, a co-manipulation activity where the robot assists the humans in reproducing simple rehabilitation tasks is considered in this paper as a case study. A custom phase law is designed and the system passivity and stability analyses are carried out. The conclusions drawn through the system stability analysis are validated by the proposed experimental results.

翻译：当需要基于参考运动定义和调整机器人任务时，动态运动基元（DMP）是一种编码参考运动的标准高效方法。标称轨迹通常通过演示编程（PbD）方法获得，即通过动觉引导教会机器人执行特定任务。随后，机械臂将再现该运动的几何路径与时间律。调整执行持续时间的基本方法涉及修改表征模型的时间常数。与之相反，本文的目标是通过一种新的空间采样算法，实现任务几何信息与支配执行的时间律的完全解耦。这催生了称为几何动态运动基元（GDMP）的新概念，其具备无相位特性，因为相位变量不再受限于演示时间律。GDMP为多种应用开辟了道路，包括受速度和加速度约束的任务时长优化，以及协同操作任务中人在回路的应用。针对后一种应用，本文以机器人辅助人类完成简单康复任务的协同操作活动作为案例研究。设计了自定义相位律，并进行了系统无源性与稳定性分析。通过系统稳定性分析得出的结论已由所提出的实验结果验证。