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Real-Time Generation of Near-Minimum-Energy Trajectories via Constraint-Informed Residual Learning

Domenico Dona', Giovanni Franzese, Cosimo Della Santina, Paolo Boscariol, Basilio Lenzo

TL;DR

The paper tackles real-time generation of near-minimum-energy trajectories for manipulators by learning only the residual needed to convert a standard prior solution into an optimal one, with hard boundary-condition enforcement. It introduces a residual-learning paradigm using either neural network ensembles or Gaussian Processes, augmented by active learning to iteratively improve data efficiency. Experiments on a pendulum, SCARA, and UR5e demonstrate substantial energy savings, feasibility, and real-time performance, with GP methods offering flexible active-learning integration and NN ensembles delivering strong baseline gains. The approach promises practical RT deployment for energy-aware robotic planning while enabling principled uncertainty-driven data augmentation and future extensions to more constraints.

Abstract

Industrial robotics demands significant energy to operate, making energy-reduction methodologies increasingly important. Strategies for planning minimum-energy trajectories typically involve solving nonlinear optimal control problems (OCPs), which rarely cope with real-time requirements. In this paper, we propose a paradigm for generating near minimum-energy trajectories for manipulators by learning from optimal solutions. Our paradigm leverages a residual learning approach, which embeds boundary conditions while focusing on learning only the adjustments needed to steer a standard solution to an optimal one. Compared to a computationally expensive OCP-based planner, our paradigm achieves 87.3% of the performance near the training dataset and 50.8% far from the dataset, while being two to three orders of magnitude faster.

Real-Time Generation of Near-Minimum-Energy Trajectories via Constraint-Informed Residual Learning

TL;DR

The paper tackles real-time generation of near-minimum-energy trajectories for manipulators by learning only the residual needed to convert a standard prior solution into an optimal one, with hard boundary-condition enforcement. It introduces a residual-learning paradigm using either neural network ensembles or Gaussian Processes, augmented by active learning to iteratively improve data efficiency. Experiments on a pendulum, SCARA, and UR5e demonstrate substantial energy savings, feasibility, and real-time performance, with GP methods offering flexible active-learning integration and NN ensembles delivering strong baseline gains. The approach promises practical RT deployment for energy-aware robotic planning while enabling principled uncertainty-driven data augmentation and future extensions to more constraints.

Abstract

Industrial robotics demands significant energy to operate, making energy-reduction methodologies increasingly important. Strategies for planning minimum-energy trajectories typically involve solving nonlinear optimal control problems (OCPs), which rarely cope with real-time requirements. In this paper, we propose a paradigm for generating near minimum-energy trajectories for manipulators by learning from optimal solutions. Our paradigm leverages a residual learning approach, which embeds boundary conditions while focusing on learning only the adjustments needed to steer a standard solution to an optimal one. Compared to a computationally expensive OCP-based planner, our paradigm achieves 87.3% of the performance near the training dataset and 50.8% far from the dataset, while being two to three orders of magnitude faster.
Paper Structure (19 sections, 2 theorems, 20 equations, 11 figures, 4 tables)

This paper contains 19 sections, 2 theorems, 20 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. METHOD
  4. Optimal Control Problem formulation
  5. Regressor design
  6. Residual Learning of explicit solutions
  7. Boundary Conditions Satisfaction
  8. Active Learning
  9. RESULTS
  10. Datasets
  11. Regressors settings
  12. Discussion
  13. Ablation study
  14. Sampling effect
  15. Energy saving capabilities
  16. ...and 4 more sections

Key Result

Lemma V.1

The derivatives of the symmetric function eq:customkernel$k_{10}$ and $k_{11}$ are both zero for $\xi = 0$ and $\xi = 1$.

Figures (11)

  • Figure 1: Schematic of the proposed residual learning paradigm. Offline, a dataset of residuals is generated and used to train a specifically designed probabilistic regressor that embeds the boundary conditions.
  • Figure 2: The proposed planner: the trained probabilistic regressor is deployed online by summing its output with the standard planner output. Moreover, the regressor outputs the epistemic uncertainty of the solution, that can be used as a measure of the lack of knowledge (data).
  • Figure 3: Proposed neural network scheme. The scaling function is applied element-wise.
  • Figure 4: Active learning approach: the uncertainty of the solution is used to generate new data.
  • Figure 5: Example trajectories of the NN and GP naive models. The violations of the boundary conditions are evident.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Lemma V.1
  • proof
  • Theorem V.1
  • proof