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Confidence-Based Skill Reproduction Through Perturbation Analysis

Brendan Hertel, S. Reza Ahmadzadeh

TL;DR

This work introduces a convex elastic-map formulation for Learning from Demonstration, enabling trajectory reproductions with a tunable confidence metric derived from perturbation analysis and Lagrangian duality. By framing the LfD problem as a constrained convex optimization with elastic-map energies, the authors derive primal and dual forms and show how constraint perturbations reveal the sensitivity of the optimal solution, which in turn defines confidence levels. The method supports constraint pruning by removing inactive constraints (where the dual variable is zero) and allows adjustable trade-offs between smoothness and constraint satisfaction, demonstrated through simulated via-point and obstacle tasks as well as a real-world door-opening with a Kinova Jaco2. The results suggest practical benefits for robust, explainable skill reproduction under varying constraints, with potential for online adaptation and handling non-stationary environments in future work.

Abstract

Several methods exist for teaching robots, with one of the most prominent being Learning from Demonstration (LfD). Many LfD representations can be formulated as constrained optimization problems. We propose a novel convex formulation of the LfD problem represented as elastic maps, which models reproductions as a series of connected springs. Relying on the properties of strong duality and perturbation analysis of the constrained optimization problem, we create a confidence metric. Our method allows the demonstrated skill to be reproduced with varying confidence level yielding different levels of smoothness and flexibility. Our confidence-based method provides reproductions of the skill that perform better for a given set of constraints. By analyzing the constraints, our method can also remove unnecessary constraints. We validate our approach using several simulated and real-world experiments using a Jaco2 7DOF manipulator arm.

Confidence-Based Skill Reproduction Through Perturbation Analysis

TL;DR

This work introduces a convex elastic-map formulation for Learning from Demonstration, enabling trajectory reproductions with a tunable confidence metric derived from perturbation analysis and Lagrangian duality. By framing the LfD problem as a constrained convex optimization with elastic-map energies, the authors derive primal and dual forms and show how constraint perturbations reveal the sensitivity of the optimal solution, which in turn defines confidence levels. The method supports constraint pruning by removing inactive constraints (where the dual variable is zero) and allows adjustable trade-offs between smoothness and constraint satisfaction, demonstrated through simulated via-point and obstacle tasks as well as a real-world door-opening with a Kinova Jaco2. The results suggest practical benefits for robust, explainable skill reproduction under varying constraints, with potential for online adaptation and handling non-stationary environments in future work.

Abstract

Several methods exist for teaching robots, with one of the most prominent being Learning from Demonstration (LfD). Many LfD representations can be formulated as constrained optimization problems. We propose a novel convex formulation of the LfD problem represented as elastic maps, which models reproductions as a series of connected springs. Relying on the properties of strong duality and perturbation analysis of the constrained optimization problem, we create a confidence metric. Our method allows the demonstrated skill to be reproduced with varying confidence level yielding different levels of smoothness and flexibility. Our confidence-based method provides reproductions of the skill that perform better for a given set of constraints. By analyzing the constraints, our method can also remove unnecessary constraints. We validate our approach using several simulated and real-world experiments using a Jaco2 7DOF manipulator arm.
Paper Structure (18 sections, 12 equations, 5 figures)

This paper contains 18 sections, 12 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Elastic Maps
  5. Elastic Maps for Trajectory Reproduction
  6. Methodology
  7. Problem Formulation: Derivation of the Primal
  8. Problem Formulation: Derivation of the Dual
  9. Interpretations
  10. Perturbation Analysis
  11. Constraints Formulation
  12. Confidence Measurement
  13. Experiments
  14. Experimental Setup
  15. Perturbing a Via-Point Constraint
  16. ...and 3 more sections

Figures (5)

  • Figure 1: Reproduction of a door opening task using a level of confidence which results in success.
  • Figure 2: Demonstration and reproductions with different confidence factors of opening a real-world box. As confidence in a reproduction increases, the constraints tighten. Left: a low-confidence reproduction does not successfully open the box. Center and right: higher confidence reproductions successfully complete the task with different features.
  • Figure 3: The perturbation analysis process shown in a simulated environment over a via-point constraint, with several reproductions of varying confidence calculated. Left: solution to the original problem with endpoint and via-point constraints. Center: perturbation analysis of the via-point constraint. The optimal value decreases when the constraint is loosened, leading to a smoother but less confident reproduction. Right: several reproductions of varying confidence levels. Confidence is shown with opacity.
  • Figure 4: Left: optimal reproduction for an obstacle avoidance task with (red) and without (blue) constraints for obstacle avoidance. Right: reproductions of varying levels of confidence, where confidence is shown with opacity.
  • Figure 5: Demonstration and reproductions with different confidence factors of opening a real-world box. Individual x and y dimensions are shown to highlight differences in reproductions. Higher confidence factors generalize less, but are more confident in the ability for the reproduction to successfully complete the skill.