Imitation learning is an intuitive approach for teaching motion to robotic systems. Although previous studies have proposed various methods to model demonstrated movement primitives, one of the limitations of existing methods is that the shape of the trajectories are encoded in high dimensional space. The high dimensionality of the trajectory representation can be a bottleneck in the subsequent process such as planning a sequence of primitive motions. We address this problem by learning the latent space of the robot trajectory. If the latent variable of the trajectories can be learned, it can be used to tune the trajectory in an intuitive manner even when the user is not an expert. We propose a framework for modeling demonstrated trajectories with a neural network that learns the low-dimensional latent space. Our neural network structure is built on the variational autoencoder (VAE) with discrete and continuous latent variables. We extend the structure of the existing VAE to obtain the decoder that is conditioned on the goal position of the trajectory for generalization to different goal positions. Although the inference performed by VAE is not accurate, the positioning error at the generalized goal position can be reduced to less than 1~mm by incorporating the projection onto the solution space. To cope with requirement of the massive training data, we use a trajectory augmentation technique inspired by the data augmentation commonly used in the computer vision community. In the proposed framework, the latent variables that encodes the multiple types of trajectories are learned in an unsupervised manner, although existing methods usually require label information to model diverse behaviors. The learned decoder can be used as a motion planner in which the user can specify the goal position and the trajectory types by setting the latent variables.

8 pages, SN Computer Science