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.

该研究介绍了一种名为 "图解教学" 的机器人获得新技能的范式，通过用户在场景图像上提供二维示意图来控制机器人的运动，并通过应用一种可微分且可逆函数（diffeomorphism）优化模型，实现对机器人运动的精确控制。实验证明，在模拟和实体机器人上，图解教学系统能够准确地教授复杂的循环运动模式。

学习轨道稳定系统以图示方式教学

Learning Orbitally Stable Systems for Diagrammatically Teaching

Learning for Demonstration (LfD) enables robots to acquire new skills by
imitating expert demonstrations, allowing users to communicate their
instructions in an intuitive manner. Recent progress in LfD often relies on
kinesthetic teaching or teleoperation as the medium for users to specify the
demonstrations. Kinesthetic teaching requires physical handling of the robot,
while teleoperation demands proficiency with additional hardware. This paper
introduces an alternative paradigm for LfD called Diagrammatic Teaching.
Diagrammatic Teaching aims to teach robots novel skills by prompting the user
to sketch out demonstration trajectories on 2D images of the scene, these are
then synthesised as a generative model of motion trajectories in 3D task space.
Additionally, we present the Ray-tracing Probabilistic Trajectory Learning
(RPTL) framework for Diagrammatic Teaching. RPTL extracts time-varying
probability densities from the 2D sketches, applies ray-tracing to find
corresponding regions in 3D Cartesian space, and fits a probabilistic model of
motion trajectories to these regions. New motion trajectories, which mimic
those sketched by the user, can then be generated from the probabilistic model.
We empirically validate our framework both in simulation and on real robots,
which include a fixed-base manipulator and a quadruped-mounted manipulator.

通过对 2D 场景图像进行示范轨迹草图，在三维任务空间中生成运动轨迹的概率模型，以此作为一种新型的 LfD（Learning for Demonstration）范式。