Programming a robot manipulator should be as intuitive as possible. To achieve that, the paradigm of teaching motion skills by providing few demonstrations has become widely popular in recent years. Probabilistic versions thereof take into account the uncertainty given by the distribution of the training data. However, precise execution of start-, via-, and end-poses at given times can not always be guaranteed. This limits the technology transfer to industrial application. To address this problem, we propose a novel constrained formulation of the Expectation Maximization algorithm for learning Gaussian Mixture Models (GMM) on Riemannian Manifolds. Our approach applies to probabilistic imitation learning and extends also to the well-established TP-GMM framework with Task-Parameterization. It allows to prescribe end-effector poses at defined execution times, for instance for precise pick & place scenarios. The probabilistic approach is compared with state-of-the-art learning-from-demonstration methods using the KUKA LBR iiwa robot. The reader is encouraged to watch the accompanying video available at https://youtu.be/JMI1YxtN9C0

翻译：编程机器人操作臂应尽可能直观。为此，通过少量示教学习运动技能的范式近年来已广泛流行。其概率版本考虑了训练数据分布带来的不确定性，然而，在特定时刻精确执行起始姿态、中间路径姿态和终止姿态并非总能得到保证，这限制了该技术向工业应用的转化。针对此问题，我们提出了一种用于黎曼流形上高斯混合模型（GMM）学习的期望最大化算法的新型约束形式。该方法适用于概率模仿学习，并可扩展至成熟的任务参数化TP-GMM框架。它允许在定义的执行时间预设末端执行器姿态，例如用于精确的抓取与放置场景。该概率方法与基于KUKA LBR iiwa机器人示教学习的最新方法进行了对比。建议读者观看配套视频：https://youtu.be/JMI1YxtN9C0