Diagrammatic Teaching is a paradigm for robots to acquire novel skills, whereby the user provides 2D sketches over images of the scene to shape the robot's motion. In this work, we tackle the problem of teaching a robot to approach a surface and then follow cyclic motion on it, where the cycle of the motion can be arbitrarily specified by a single user-provided sketch over an image from the robot's camera. Accordingly, we introduce the \emph{Stable Diffeomorphic Diagrammatic Teaching} (SDDT) framework. SDDT models the robot's motion as an \emph{Orbitally Asymptotically Stable} (O.A.S.) dynamical system that learns to follow the user-specified sketch. This is achieved by applying a \emph{diffeomorphism}, i.e. a differentiable and invertible function, to morph a known O.A.S. system. The parameterised diffeomorphism is then optimised with respect to the Hausdorff distance between the limit cycle of our modelled system and the sketch, to produce the desired robot motion. We provide theoretical insight into the behaviour of the optimised system and also empirically evaluate SDDT, both in simulation and on a quadruped with a mounted 6-DOF manipulator. Results show that we can diagrammatically teach complex cyclic motion patterns with a high degree of accuracy.

翻译：图解式教学是一种让机器人获取新技能的范式，用户通过场景图像上的二维草图来塑造机器人运动。本文研究如何教会机器人接近表面并在其上执行循环运动，其中运动周期可由用户通过机器人相机图像上的单个草图任意指定。为此，我们提出了稳定微分同胚图解式教学（SDDT）框架。SDDT将机器人运动建模为一个轨道渐近稳定（O.A.S.）动力系统，该系统学习跟随用户指定的草图。这是通过应用微分同胚（即一个可微且可逆的函数）来变形已知的O.A.S.系统实现的。随后，基于建模系统的极限环与草图之间的豪斯多夫距离对参数化微分同胚进行优化，以生成所需的机器人运动。我们提供了优化系统行为的理论见解，并在仿真环境和配备六自由度机械臂的四足机器人上对SDDT进行了实证评估。结果表明，我们能够以高精度图解式教授复杂的循环运动模式。