Gaussian process dynamical models over dual quaternions

This paper presents a method for learning nonlinear rigid body dynamics in the special Euclidean group SE(3). The method is based on the Gaussian process dynamical model (GPDM), which combines two Gaussian processes (GPs), one for representing unknown dynamics in a space R^d with reduced dimensionality and the other for transforming the reduced space back to the state space of the high dimensional measurements R^D. We introduce in this paper an enhanced GPDM, which extends the dynamics modeling space from Euclidean space to the special Euclidean group SE(3). This allows for accurate modeling of unknown dynamics incorporating rotation and translation. Therefore, the unknown dynamics are described by a GP over dual quaternions, denoted by GPH_D, which extends the state of the art GP to a non-Euclidean input space SE(3). Further, we provide a proof that the squared exponential kernel used in the GPH_D defines a valid covariance function. In conclusion we illustrate how the GPDM over dual quaternions outperforms the traditional GPDM depending on the amount of training data and rotation magnitude.