Arc-Length-Based Warping for Robot Skill Synthesis from Multiple Demonstrations

TL;DR

The paper tackles temporal distortions in Learning from Demonstration by introducing Spatial Sampling (SS), a method that computes arc-length parametrizations of robot trajectories to achieve time-independent alignment. It extends SS to handle multiple demonstrations, provides a principled Δ selection procedure based on Hausdorff distance to control approximation accuracy, and analyzes computational cost. Through experiments on LASA handwriting data and a custom geometric dataset, SS demonstrates improved trajectory alignment and barycenter quality, often outperforming DTW-based baselines, especially when integrated with GMR or DTAN. The approach offers a robust, interpretable, and scalable alternative for skill synthesis from demonstrations and is accompanied by an open-source dataset for the community.

Abstract

In robotics, Learning from Demonstration (LfD) aims to transfer skills to robots by using multiple demonstrations of the same task. These demonstrations are recorded and processed to extract a consistent skill representation. This process typically requires temporal alignment through techniques such as Dynamic Time Warping (DTW). In this paper, we consider a novel algorithm, named Spatial Sampling (SS), specifically designed for robot trajectories, that enables time-independent alignment of the trajectories by providing an arc-length parametrization of the signals. This approach eliminates the need for temporal alignment, enhancing the accuracy and robustness of skill representation, especially when recorded movements are subject to intermittent motions or extremely variable speeds, a common characteristic of operations based on kinesthetic teaching, where the operator may encounter difficulties in guiding the end-effector smoothly. To prove this, we built a custom publicly available dataset of robot recordings to test real-world movements, where the user tracks the same geometric path multiple times, with motion laws that vary greatly and are subject to starting and stopping. The SS demonstrates better performances against state-of-the-art algorithms in terms of (i) trajectory synchronization and (ii) quality of the extracted skill.

Figures (14)

• Figure 1: Learning from Demonstration (LfD) framework (a) and approximation accuracy of the computed barycenter (red) with varying number of demonstrations $D$ (black) (b).
• Figure 2: Working principle of the Spatial Sampling (SS) algorithm for two different $\Delta$ values. Lower $\Delta$ induces a better approximation of the recorded curve ${\boldsymbol r}$. Note that the condition\ref{['eq:norm_Delta1']} focuses on a spatial constraint, allowing the SS algorithm to be independent on the demonstration's timing.
• Figure 3: Numerical example of Sec. \ref{['subsubsec:Numerical example']}. (a) Result of the Optimization Problem \ref{['opt:optim_delta']} applied to the trajectory shown in (b) using $d_H^*=0.025$ and $d_{H,t}=0.1\cdot 10^{-3}$. (b) Linear interpolation ${\boldsymbol r}_L(t)$ of the samples of the original trajectory, samples ${\boldsymbol x}_\Delta({\boldsymbol s}_k)$ of the filtered trajectory using the optimal $\Delta^\star=0.0256$, and their linear interpolation ${\boldsymbol r}_L(t_k)$.
• Figure 4: Box plots and table of simulation results on LASA handwriting dataset khansari2011learning when comparing time (blue) versus arc-length (green) parametrization obtained through SS. From the left, the box reports the following metrics: Rho ($\rho$), Entropy ($H$), Sum-Normalized Cross-Spectral Density (snCSD), $\ell_2$ norm error.
• Figure 6: Results of the $d_H$ and $d_{DTW}$ metrics (a) for the DTW/GMR combination across reference variation and (b), for TIME/GMR, DTW/GMR and SS/GMR when varying the number of components $G$ in the GMM model.