# Constrained Equation Learner Networks for Precision-Preserving Extrapolation of Robotic Skills

arXiv: https://arxiv.org/abs/2311.02475 · HTML: https://mlsanity.com/paper/2311.02475/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 45 references that are in the map.
- First posted on arXiv 4 Nov 2023.

## Builds on (most cited first, 14 of 14)

- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [Kernelized Movement Primitives](https://mlsanity.com/paper/1708.08638/) (2017)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization](https://mlsanity.com/paper/2108.00069/) (2021)
- [Learning Barrier Functions for Constrained Motion Planning with Dynamical Systems](https://mlsanity.com/paper/2003.11500/) (2020)
- [Towards Orientation Learning and Adaptation in Cartesian Space](https://mlsanity.com/paper/1907.03918/) (2019)
- [Constrained Probabilistic Movement Primitives for Robot Trajectory Adaptation](https://mlsanity.com/paper/2101.12561/) (2021)
- [EKMP: Generalized Imitation Learning with Adaptation, Nonlinear Hard Constraints and Obstacle Avoidance](https://mlsanity.com/paper/2103.00452/) (2021)
- [Learning and Extrapolation of Robotic Skills using Task-Parameterized Equation Learner Networks](https://mlsanity.com/paper/2212.08531/) (2022)
- [A Library for Constraint Consistent Learning](https://mlsanity.com/paper/1807.04676/) (2018)

## Abstract (arXiv)

In Programming by Demonstration, the robot learns novel skills from human demonstrations. After learning, the robot should be able not only to reproduce the skill, but also to generalize it to shifted domains without collecting new training data. Adaptation to similar domains has been investigated in the literature; however, an open problem is how to adapt learned skills to different conditions that are outside of the data distribution, and, more important, how to preserve the precision of the desired adaptations. This paper presents a novel supervised learning framework called Constrained Equation Learner Networks that addresses the trajectory adaptation problem in Programming by Demonstrations from a constrained regression perspective. While conventional approaches for constrained regression use one kind of basis function, e.g., Gaussian, we exploit Equation Learner Networks to learn a set of analytical expressions and use them as basis functions. These basis functions are learned from demonstration with the objective to minimize deviations from the training data while imposing constraints that represent the desired adaptations, like new initial or final points or maintaining the trajectory within given bounds. Our approach addresses three main difficulties in adapting robotic trajectories: 1) minimizing the distortion of the trajectory for new adaptations; 2) preserving the precision of the adaptations; and 3) dealing with the lack of intuition about the structure of basis functions. We validate our approach both in simulation and in real experiments in a set of robotic tasks that require adaptation due to changes in the environment, and we compare obtained results with two existing approaches. Performed experiments show that Constrained Equation Learner Networks outperform state of the art approaches by increasing generalization and adaptability of robotic skills.