Robot Learning from Demonstration Using Elastic Maps

TL;DR

The paper presents elastic maps as a convex, optimization-based framework for Learning from Demonstration, encoding robot trajectories as node-spring graphs and fitting them via a three-term energy: $U_Y$ (data fit), $U_E$ (edge stretching), and $U_R$ (bending). By leveraging an EM-based optimization and configurable initializations and weightings, the method can reproduce single or multiple demonstrations, enforce start/end/via-point constraints, and preserve curvature and smoothness. Empirical results across simulated and real-world tasks (including a UR5e manipulator) show competitive performance against DMPs and GMM/GMR baselines, with robust constraint handling and favorable speed. The approach offers a flexible, interpretable alternative for trajectory learning that readily integrates constraints and scales to varied demonstrations and tasks.

Abstract

Learning from Demonstration (LfD) is a popular method of reproducing and generalizing robot skills from human-provided demonstrations. In this paper, we propose a novel optimization-based LfD method that encodes demonstrations as elastic maps. An elastic map is a graph of nodes connected through a mesh of springs. We build a skill model by fitting an elastic map to the set of demonstrations. The formulated optimization problem in our approach includes three objectives with natural and physical interpretations. The main term rewards the mean squared error in the Cartesian coordinate. The second term penalizes the non-equidistant distribution of points resulting in the optimum total length of the trajectory. The third term rewards smoothness while penalizing nonlinearity. These quadratic objectives form a convex problem that can be solved efficiently with local optimizers. We examine nine methods for constructing and weighting the elastic maps and study their performance in robotic tasks. We also evaluate the proposed method in several simulated and real-world experiments using a UR5e manipulator arm, and compare it to other LfD approaches to demonstrate its benefits and flexibility across a variety of metrics.

Figures (9)

• Figure 1: An elastic map reproduction of the pressing skill on a UR5e robot. The pressing points are constrained as via-points.
• Figure 2: Visualization of a simple polyline elastic map and associated energies. This map includes: 3 nodes $y_1$, $y_2$, and $y_3$ with energy $U_Y$; 2 edges $e_1$={$y_1$, $y_2\}$ and $e_2$={$y_2$, $y_3\}$ with energy $U_E$; and 1 rib $r_1$={$y_1$, $y_2$, $y_3$} with energy $U_R$.
• Figure 3: Three downsampling methods examined for initialization of elastic maps on a 2D handwriting demonstration.
• Figure 4: Elastic map reproductions ($\color{red} \bm{-}$) with different numbers of demonstrations ($\color{gray} \bm{-}$) and constraints ($\bullet$) for various handwriting shapes.
• Figure 5: Effect of the number of nodes $N$ on computation time ($\color{bblue} \bm{-}$) and dissimilarity ($\color{red} \bm{-}$). Note the log scale on the dissimilarity axis.