  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.
