  Movement Primitives (MPs) are a well-known concept to represent and generate
modular trajectories. MPs can be broadly categorized into two types: (a)
dynamics-based approaches that generate smooth trajectories from any initial
state, e. g., Dynamic Movement Primitives (DMPs), and (b) probabilistic
approaches that capture higher-order statistics of the motion, e. g.,
Probabilistic Movement Primitives (ProMPs). To date, however, there is no
method that unifies both, i. e. that can generate smooth trajectories from an
arbitrary initial state while capturing higher-order statistics. In this paper,
we introduce a unified perspective of both approaches by solving the ODE
underlying the DMPs. We convert expensive online numerical integration of DMPs
into basis functions that can be computed offline. These basis functions can be
used to represent trajectories or trajectory distributions similar to ProMPs
while maintaining all the properties of dynamical systems. Since we inherit the
properties of both methodologies, we call our proposed model Probabilistic
Dynamic Movement Primitives (ProDMPs). Additionally, we embed ProDMPs in deep
neural network architecture and propose a new cost function for efficient
end-to-end learning of higher-order trajectory statistics. To this end, we
leverage Bayesian Aggregation for non-linear iterative conditioning on sensory
inputs. Our proposed model achieves smooth trajectory generation,
goal-attractor convergence, correlation analysis, non-linear conditioning, and
online re-planing in one framework.
