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 contribute the Stable Diffeomorphic Diagrammatic Teaching (SDDT) framework. SDDT models the robot's motion as an Orbitally Asymptotically Stable (O.A.S.) dynamical system that learns to stablize based on a single diagrammatic sketch provided by the user. 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 novel 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：图示化教学是一种让机器人获取新技能的范式，用户通过在场景图像上绘制二维草图来引导机器人的运动。本研究针对机器人接近表面并执行表面上的循环运动这一任务，其中循环的周期可由用户根据机器人摄像头捕获的图像通过单一草图任意指定。为此，我们提出了稳定微分同胚图示化教学（SDDT）框架。SDDT将机器人的运动建模为轨道渐近稳定（O.A.S.）的动力学系统，该系统能根据用户提供的单一图示化草图学习稳定行为。这是通过应用一个**微分同胚**（即可微且可逆的函数）对已知的O.A.S.系统进行形变实现的。随后，我们通过最小化建模系统极限环与草图之间的豪斯多夫距离来优化参数化微分同胚，从而生成期望的机器人运动。我们为优化系统的行为提供了新颖的理论洞察，并通过仿真实验以及搭载六自由度机械臂的四足机器人平台对SDDT进行了实证评估。结果表明，该方法能够以高精度图示化地教授复杂的循环运动模式。