  Imitation learning approaches achieve good generalization within the range of
the training data, but tend to generate unpredictable motions when querying
outside this range. We present a novel approach to imitation learning with
enhanced extrapolation capabilities that exploits the so-called Equation
Learner Network (EQLN). Unlike conventional approaches, EQLNs use supervised
learning to fit a set of analytical expressions that allows them to extrapolate
beyond the range of the training data. We augment the task demonstrations with
a set of task-dependent parameters representing spatial properties of each
motion and use them to train the EQLN. At run time, the features are used to
query the Task-Parameterized Equation Learner Network (TP-EQLN) and generate
the corresponding robot trajectory. The set of features encodes kinematic
constraints of the task such as desired height or a final point to reach. We
validate the results of our approach on manipulation tasks where it is
important to preserve the shape of the motion in the extrapolation domain. Our
approach is also compared with existing state-of-the-art approaches, in
simulation and in real setups. The experimental results show that TP-EQLN can
respect the constraints of the trajectory encoded in the feature parameters,
even in the extrapolation domain, while preserving the overall shape of the
trajectory provided in the demonstrations.
